Chapter 02 - Probability.ipynb

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   "source": [
    "## 2. Probability"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "### 2.2 Sample Spaces and Events"
   ]
  },
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   "metadata": {},
   "source": [
    "The **sample space** $\\Omega$  is the set of possible outcomes of an experiment.  Points $\\omega$ in $\\Omega$ are called **sample outcomes** or **realizations**.  **Events** are subsets of $\\Omega$."
   ]
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    "Given an event $A$, let $A^c = \\{ \\omega \\in \\Omega : \\text{not } (\\omega \\in A) \\}$ denote the complement of $A$.  The complement of $\\Omega$ is the empty set $\\varnothing$.  The union of events $A$ and $B$ is defined as $A \\cup B = \\{ \\omega \\in \\Omega : \\omega \\in A \\text{ or } \\omega \\in B \\}$.  If $A_1, A_2, \\dots$ is a sequence of sets, then\n",
    "\n",
    "$$ \\cup_{i=1}^\\infty A_i = \\left\\{ \\omega \\in \\Omega : \\omega \\in A_i \\text{ for some } i \\right\\}$$\n",
    "\n",
    "The intersection of $A$ and $B$ is $A \\cap B = \\{ \\omega \\in \\Omega : \\omega \\in A \\text{ and } \\omega \\in B \\}$.  If $A_1, A_2, \\dots$ is a sequence of sets then\n",
    "\n",
    "$$ \\cap_{i=1}^\\infty A_i = \\left\\{ \\omega \\in \\Omega : \\omega \\in A_i \\text{ for all } i \\right\\}$$\n",
    "\n",
    "Let $A - B = \\left\\{ \\omega \\in \\Omega : \\omega \\in A \\text{ and not } (\\omega \\in B) \\right\\}$.  If every element of $A$ is contained in $B$ we write $A \\subset B$ or $B \\supset A$.  If $A$ is a finite set, let $|A|$ denote the number of elements in $A$."
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    "| notation           | meaning                                            |\n",
    "|--------------------|----------------------------------------------------|\n",
    "| $\\Omega$           | sample space                                       |\n",
    "| $\\omega$           | outcome                                            | \n",
    "| $A$                | event (subset of $\\Omega$)                         |\n",
    "| $\\vert A \\vert$    | number of elements in $A$ (if finite)              |\n",
    "| $A^c$              | complement of $A$ (not $A$)                        |\n",
    "| $A \\cup B$         | union ($A$ or $B$)                                 |\n",
    "| $A \\cap B$ or $AB$ | intersection ($A$ and $B$)                         |\n",
    "| $A - B$            | set difference (points in $A$ but not in $B$)      |\n",
    "| $A \\subset B$      | set inclusion ($A$ is a subset of or equal to $B$) |\n",
    "| $\\varnothing$      | null event (always false)                          |\n",
    "| $\\Omega$           | true event (always true)                           |"
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   "source": [
    "We say that $A_1, A_2, \\dots$ are **disjoint** or **mutually exclusive** if $A_i \\cap A_j = \\varnothing$ whenever $i \\neq j$.  A **partition** of $\\Omega$ is a sequence of disjoint sets $A_1, A_2, \\dots$ such that $\\cup_{i=1}^\\infty A_i = \\Omega$.  Given an event $A$, define the **indicator function of $A$** by\n",
    "\n",
    "$$ I_A(\\omega) = I(\\omega \\in A) = \\begin{cases}\n",
    "1 &\\text{if } \\omega \\in A \\\\\n",
    "0 &\\text{otherwise}\n",
    "\\end{cases}$$\n",
    "\n",
    "A sequence of sets $A_1, A_2, \\dots$ is **monotone increasing** if $A_1 \\subset A_2 \\subset \\dots$, and we define $\\lim_{n \\rightarrow \\infty} A_n = \\cup_{i=1}^\\infty A_i$.  A sequence of sets $A_1, A_2, \\dots$ is **monotone decreasing** if $A_1 \\supset A_2 \\supset \\dots$ and then we define $\\lim_{n \\rightarrow \\infty} A_n = \\cap_{i=1}^n A_i$.  In either case, we will write $A_n \\rightarrow A$."
   ]
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   "source": [
    "### 2.3 Probability"
   ]
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   "source": [
    "A function $\\mathbb{P}$ that assign a real number $\\mathbb{P}(A)$ to each event $A$  is a **probability distribution** or a **probability measure** if it satisfies the following three axioms:\n",
    "\n",
    "- **Axiom 1**: $\\mathbb{P}(A) \\geq 0$ for every $A$\n",
    "- **Axiom 2**: $\\mathbb{P}(\\Omega) = 1$\n",
    "- **Axiom 3**: If $A_1, A_2, \\dots$ are disjoint then\n",
    "\n",
    "$$ \\mathbb{P} \\left( \\cup_{i=1}^\\infty A_i \\right) = \\sum_{i=1}^\\infty \\mathbb{P}(A_i) $$"
   ]
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    "A few properties that can be derived from the axioms:\n",
    "\n",
    "- $\\mathbb{P}(\\varnothing) = 0$\n",
    "- $A \\subset B \\Rightarrow \\mathbb{P}(A) \\leq \\mathbb{P}(B)$\n",
    "- $0 \\leq \\mathbb{P}(A) \\leq 1$\n",
    "- $\\mathbb{P}\\left(A^c\\right) = 1 - \\mathbb{P}(A)$\n",
    "- $A \\cap B = \\varnothing \\Rightarrow \\mathbb{P}(A \\cup B) = \\mathbb{P}(A) + \\mathbb{P}(B)$"
   ]
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   "source": [
    "**Lemma 2.6**.  For any events $A$ and $B$, $\\mathbb{P}(A \\cup B) = \\mathbb{P}(A) + \\mathbb(B) - \\mathbb{P}(AB)$."
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    "**Proof**. \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbb{P}(A \\cup B) &= \\mathbb{P}\\left( (AB^c) \\cup (AB) \\cup (A^cB) \\right) \\\\\n",
    "&= \\mathbb{P}(AB^c) + \\mathbb{P}(AB) + \\mathbb{P}(A^cB) \\\\\n",
    "&= \\mathbb{P}(AB^c) + \\mathbb{P}(AB) + \\mathbb{P}(A^cB) + \\mathbb{P}(AB) - \\mathbb{P}(AB) \\\\\n",
    "&= \\mathbb{P}((AB^c) \\cup (AB)) + \\mathbb{P}((A^cB) \\cup (AB)) - \\mathbb{P}(AB) \\\\\n",
    "&= \\mathbb{P}(A) + \\mathbb{P}(B) - \\mathbb{P}(AB)\n",
    "\\end{align}\n",
    "$$"
   ]
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   "source": [
    "**Theorem 2.8 (Continuity of Probabilities)**.  If $A_n \\rightarrow A$ then $\\mathbb{P}(A_n) \\rightarrow \\mathbb{P}(A)$ as $n \\rightarrow \\infty$."
   ]
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   "source": [
    "**Proof**.  Suppose that $A_n$ is monotone increasing, $A_1 \\subset A_2 \\subset \\dots$.  Let $B_1 = A_1$, and $B_{n+1} = A_{n+1} - A_n$ for $n > 1$.   The $B_i$'s are disjoint by construction, and $A_n = \\cup_{i=1}^n A_i = \\cup_{i=1}^n B_i$ for all $n$.  From axiom 3,\n",
    "\n",
    "$$ \\mathbb{P}(A_n) = \\mathbb{P}\\left( \\cup_{i=1}^n B_i \\right)  = \\sum_{i=1}^n \\mathbb{P}(B_i) $$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ \\lim_{n \\rightarrow \\infty} \\mathbb{P}(A_n) = \\lim_{n \\rightarrow \\infty} \\sum_{i=1}^n \\mathbb{P}(B_n) = \\sum_{i=1}^\\infty \\mathbb{P}(B_n) = \\mathbb{P}\\left( \\cup_{i=1}^\\infty B_i \\right) = \\mathbb{P}(A) $$"
   ]
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   "source": [
    "### 2.4 Probability on Finite Sample Spaces"
   ]
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   "source": [
    "If $\\Omega$ is finite and each outcome is equally likely, then \n",
    "\n",
    "$$ \\mathbb{P}(A) = \\frac{|A|}{|\\Omega|} $$\n",
    "\n",
    "which is called the **uniform probability distribution**.\n",
    "\n",
    "We will need a few facts from counting theory later.  \n",
    "\n",
    "- Given $n$ objects, the number of way or ordering these objects is $n! = n \\cdot (n - 1) \\cdot (n - 2) \\cdots 3 \\cdot 2 \\cdot 1$.  We define $0! = 1$.\n",
    "- We define\n",
    "\n",
    "  $$ \\binom{n}{k} = \\frac{n!}{k! (n - k)!} $$\n",
    "\n",
    "  read \"n choose k\", which is the number of different ways of choosing $k$ objects from $n$.\n",
    "  \n",
    "- Note that choosing a subset $k$ objects can be mapped to choosing the complement set of $n - k$ objects, so\n",
    "\n",
    "  $$ \\binom{n}{k} = \\binom{n}{n - k} $$\n",
    " \n",
    "  and that there is only one way of choosing the empty set, so\n",
    "  \n",
    "  $$ \\binom{n}{0} = \\binom{n}{n} = 1$$"
   ]
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   "metadata": {},
   "source": [
    "### 2.5 Independent Events"
   ]
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   "source": [
    "- Two events $A$ and $B$ are **independent** if\n",
    "\n",
    "    $$ \\mathbb{P}(AB) = \\mathbb{P}(A) \\mathbb{P}(B) $$\n",
    "\n",
    "    and we write $A \\text{ ⫫ } B$.  A set of events $\\{ A_i : i \\in I \\}$ is independent if\n",
    "\n",
    "    $$ \\mathbb{P} \\left( \\cap_{i \\in J} A_i \\right) = \\prod_{i \\in J} \\mathbb{P}(A_i) $$\n",
    "\n",
    "    for every finite subset $J$ of $I$.\n",
    "\n",
    "- Independence is sometimes assumed and sometimes derived.\n",
    "- Disjoint events with positive probability are not independent."
   ]
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   "source": [
    "### 2.6 Conditional Probability"
   ]
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   "source": [
    "- If $\\mathbb{P}(B) > 0$ then the **conditional probability** of $A$ given $B$ is\n",
    "\n",
    "  $$ \\mathbb{P}(A | B) = \\frac{\\mathbb{P}(AB)}{\\mathbb{P}(B)} $$\n",
    "  \n",
    "- $\\mathbb{P}(\\cdot | B)$ satisfies the axioms of probability, for fixed $B$.  In general, $\\mathbb{P}(A | \\cdot)$ does **not** satisfies the axioms of probability for fixed $A$.\n",
    "\n",
    "- In general, $\\mathbb{P}(B | A) \\neq \\mathbb{P}(A | B)$.\n",
    "- $A$ and $B$ are independent if and only if $\\mathbb{P}(A | B) = \\mathbb{P}(A)$."
   ]
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   "source": [
    "### 2.7 Bayes' Theorem"
   ]
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   "source": [
    "**Theorem 2.15 (The Law of Total Probability)**.  Let $A_1, \\dots, A_k$ be a partition of $\\Omega$.  Then, for any event $B$,\n",
    "\n",
    "$$ \\mathbb{P}(B) = \\sum_{i=1}^k \\mathbb{P}(B | A_i) \\mathbb{P}(A_i) $$"
   ]
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   "source": [
    "**Proof**.  Let $C_j = BA_j$.  Note that the $C_j$'s are disjoint and that $B = \\cup_{i=1}^k C_j$.  Hence\n",
    "\n",
    "$$ \\mathbb{P}(B) = \\sum_j \\mathbb{P}(C_j)  = \\sum_j \\mathbb{P}(BA_j) = \\sum_j \\mathbb{P}(B | A_j) \\mathbb{P}(A_j) $$"
   ]
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   "source": [
    "**Theorem 2.16 (Bayes' Theorem)**.  Let $A_1, \\dots, A_k$ be a partition of $\\Omega$ such that $\\mathbb{P}(A_i) > 0$ for each $i$.  If $\\mathbb{P}(B) > 0$, then, for each $i = 1, \\dots, k$,\n",
    "\n",
    "$$ \\mathbb{P}(A_i | B) = \\frac{\\mathbb{P}(B | A_i) \\mathbb{P}(A_i)}{\\sum_j \\mathbb{P}(B | A_j) \\mathbb{P}(A_j)} $$\n",
    "\n",
    "We call $\\mathbb{P}(A_i)$ the **prior probability** of $A_i$ and $\\mathbb{P}(A_i | B)$ the **posterior probability** of $A_i$."
   ]
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   "source": [
    "**Proof**.  We apply the definition of conditional probability twice, followed by the law of total probability:\n",
    "\n",
    "$$ \\mathbb{P}(A_i | B) = \\frac{\\mathbb{P}(A_i B) }{\\mathbb{P}(B)} = \\frac{\\mathbb{P}(B | A_i) \\mathbb{P}(A_i)}{\\mathbb{P}(B)} = \\frac{\\mathbb{P}(B | A_i) \\mathbb{P}(A_i)}{\\sum_j \\mathbb{P}(B | A_j) \\mathbb{P}(A_j)}$$"
   ]
  },
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   "metadata": {},
   "source": [
    "### 2.9 Technical Appendix"
   ]
  },
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   "metadata": {},
   "source": [
    "Generally, it is not feasible to assign probabilities to all the subsets of a sample space $\\Omega$.  Instead, one restricts attention to a set of events called a **$\\sigma$-algebra** or a **$\\sigma$-field**, which is a class $\\mathcal{A}$ that satisfies:\n",
    "\n",
    "- $\\varnothing \\in \\mathcal{A}$\n",
    "- If $A_1, A_2, \\dots \\in \\mathcal{A}$, then $\\cup_{i=1}^\\infty A_i \\in \\mathcal{A}$\n",
    "- $A \\in \\mathcal{A}$ implies that $A^c \\in \\mathcal{A}$\n",
    "\n",
    "The sets in $\\mathcal{A}$ are said to be **measurable**.  We call $(\\Omega, \\mathcal{A})$ a **measurable space**.  If $\\mathbb{P}$ is a probability measure defined in $\\mathcal{A}$ then $(\\Omega, \\mathcal{A}, \\mathbb{P})$ is a **probability space**.  When $\\Omega$ is the real line, we take $\\mathcal{A}$ to be the smallest $\\sigma$-field that contains all off the open sets, which is called the **Borel $\\sigma$-field**."
   ]
  },
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   "metadata": {},
   "source": [
    "### 2.10 Exercises"
   ]
  },
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   "metadata": {},
   "source": [
    "**Exercise 2.10.1**.  Fill in the details in the proof of Theorem 2.8.  Also, prove the monotone decreasing case.\n",
    "\n",
    "If $A_n \\rightarrow A$ then $\\mathbb{P}(A_n) \\rightarrow \\mathbb{P}(A)$ as $n \\rightarrow \\infty$."
   ]
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    "**Solution**.  \n",
    "\n",
    "Suppose that $A_n$ is monotone increasing, $A_1 \\subset A_2 \\subset \\dots$.  Let $B_1 = A_1$, and $B_{i+1} = A_{i+1} - A_i$ for $i > 1$.   \n",
    "\n",
    "The $B_i$'s are disjoint by construction: assuming without loss of generality $i < j$,  $\\omega \\in B_i \\cap B_j$ implies that $\\omega$ is in $A_j$, $A_i$, but not in $A_{j - 1}$, $A_{i - 1}$, where $A_0 = \\varnothing$.  In particular, this means that $\\omega \\in A_{i}$ but not $\\omega \\in A_{j - 1}$.  Since $A_i \\subset A_{j - 1}$, this implies that no such $\\omega$ can satisfy those properties, and so $B_i$ and $B_j$ are disjoint.\n",
    "\n",
    "Note that $A_n = \\cup_{i=1}^n A_i = \\cup_{i=1}^n B_i$ for all $n$:\n",
    "\n",
    "$$ \\cup_{i=1}^n B_i = \\cup_{i=1}^n (A_i - A_{i - 1}) \\subset \\cup_{i=1}^n A_i = A_n $$\n",
    "\n",
    "Also note that $A_n \\subset \\cup_{i=1}^n B_i$, since, if $f(\\omega) = \\min \\{ k : \\omega \\in A_k \\}$, then $\\omega \\in B_{f(\\omega)}$, so all elements of $A_n$ are in some $B_k$.\n",
    "\n",
    "The proof follows as given; from axiom 3,\n",
    "\n",
    "$$ \\mathbb{P}(A_n) = \\mathbb{P}\\left( \\cup_{i=1}^n B_i \\right) = \\sum_{i=1}^n \\mathbb{P}(B_i) $$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ \\lim_{n \\rightarrow \\infty} \\mathbb{P}(A_n) = \\lim_{n \\rightarrow \\infty} \\sum_{i=1}^n \\mathbb{P}(B_n) = \\sum_{i=1}^\\infty \\mathbb{P}(B_n) = \\mathbb{P}\\left( \\cup_{i=1}^\\infty B_i \\right) = \\mathbb{P}(A) $$\n",
    "\n",
    "The monotone decreasing case can be obtained by looking at the complementary series $A_1^c, A_2^c, \\dots$,which is monotone increasing.  We get\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\lim_{n \\rightarrow \\infty} \\mathbb{P}(A_n^c) &= \\mathbb{P}(A^c) \\\\\n",
    "\\lim_{n \\rightarrow \\infty} 1 - \\mathbb{P}(A_n^c) &= 1 - \\mathbb{P}(A^c) \\\\\n",
    "\\lim_{n \\rightarrow \\infty} \\mathbb{P}(A_n) &= \\mathbb{P}(A)\n",
    "\\end{align}\n",
    "$$\n"
   ]
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   "source": [
    "**Exercise 2.10.2**.  Prove the statements in equation (2.1).\n",
    "\n",
    "- $\\mathbb{P}(\\varnothing) = 0$\n",
    "- $A \\subset B \\Rightarrow \\mathbb{P}(A) \\leq \\mathbb{P}(B)$\n",
    "- $0 \\leq \\mathbb{P}(A) \\leq 1$\n",
    "- $\\mathbb{P}\\left(A^c\\right) = 1 - \\mathbb{P}(A)$\n",
    "- $A \\cap B = \\varnothing \\Rightarrow \\mathbb{P}(A \\cup B) = \\mathbb{P}(A) + \\mathbb{P}(B)$"
   ]
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   "source": [
    "**Solution**.\n",
    "\n",
    "- By partitioning the event space $\\Omega$ into disjoint partitions $(\\Omega, \\varnothing)$ we get\n",
    "\n",
    "  $$ \\mathbb{P}(\\Omega) + \\mathbb{P}(\\varnothing) = \\mathbb{P}(\\Omega) \\Rightarrow \\mathbb{P}(\\varnothing) = 0 $$\n",
    "  \n",
    "- Assuming $A \\subset B$ and partitioning $B$ as $(A, B - A)$, we get\n",
    "\n",
    "  $$ \\mathbb{P}(A) + \\mathbb{P}(B - A) = \\mathbb{P}(B) \\Rightarrow \\mathbb{P}(A) \\leq \\mathbb{P}(B) $$\n",
    "  \n",
    "- $\\mathbb{P}(A) \\geq 0$ from axiom 1.  By partitioning $\\Omega$ as $(A, A^c)$, we get\n",
    "\n",
    "  $$ \\mathbb{P}(A) + \\mathbb{P}(A^c) = \\mathbb{P}(\\Omega) = 1 \\Rightarrow \\mathbb{P}(A) \\leq 1 $$\n",
    "  \n",
    "- By partitioning $\\Omega$ as $(A, A^c)$, we get\n",
    "\n",
    "  $$ \\mathbb{P}(A) + \\mathbb{P}(A^c) = \\mathbb{P}(\\Omega) = 1 \\Rightarrow \\mathbb{P}(A) = 1 - \\mathbb{P}(A^c) $$\n",
    "  \n",
    "- Assuming $A$, $B$ are disjoint, we partition $A \\cup B$ in $(A, B)$ and get:\n",
    "\n",
    "  $$ \\mathbb{P}(A \\cup B) = \\mathbb{P}(A) + \\mathbb{P}(B) $$"
   ]
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   "source": [
    "**Exercise 2.10.3**.  Let $\\Omega$ be a sample space and let $A_1, A_2, \\dots$ be events.  Define $B_n = \\cup_{i=n}^\\infty A_i$ and $C_n = \\cap_{i=n}^\\infty A_i$\n",
    "\n",
    "**(a)** Show that $B_1 \\supset B_2 \\supset \\cdots$ and $C_1 \\subset C_2 \\subset \\cdots$.\n",
    "\n",
    "**(b)** Show that $\\omega \\in \\cap_{n = 1}^\\infty B_n$ if and only if $\\omega$ belongs to an infinite number of the events $A_1, A_2, \\dots$.\n",
    "\n",
    "**(c)** Show that $\\omega \\in \\cup_{n = 1}^\\infty C_n$ if and only if $\\omega$ belongs to all of the events $A_1, A_2, \\dots$ except possibly a finite number of those events."
   ]
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   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**  By construction, $B_{n+1} = A_n \\cup B_n$ and so $B_{n + 1} \\supset B_n$.  Similarly, $C_{n+1} = A_n \\cap C_n$ and so $C_{n + 1} \\subset C_n$."
   ]
  },
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   "source": [
    "**(b)**\n",
    "\n",
    "- Assume $\\omega$ belongs to an infinite number of the events, $\\omega \\in A_j$ for $j \\in J(\\omega)$.  Then, for every $n$, there is a $m \\geq n$ such that $m \\in J(\\omega)$, and so $\\omega \\in B_n$ for every $n$.  This implies that $\\omega \\in \\cap_{n = 1}^\\infty B_n$.\n",
    "\n",
    "- Assume that $\\omega \\in \\cap_{n = 1}^\\infty B_n$.  Then, for every $n$, $\\omega \\in B_n$, so for every $n$ there is a $m \\geq n$ such that $\\omega \\in A_m$.  This implies there is an infinite number of such events $A_m$."
   ]
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   "source": [
    "**(c)** \n",
    "\n",
    "Let's prove the contrapositive.  \n",
    "\n",
    "- Assume that $\\omega$ does not belong to an infinite number of events $A_i$.  Then, for every $n$, there is a $m \\geq$ such that $\\omega \\in A_m^c$, and so $\\omega$ is not in $C_n$.  Since $\\omega$ is not in none of the $C_n$'s, it is not in the union of all $C_n$'s either. \n",
    "- Assume that $\\omega$ is not in the union of all $C_n$.  This implies that $\\omega$ is is not in any event $C_n$.  This implies that, for every $n$, there is a $m \\geq n$ such that $\\omega$ is not in $A_m$.  This implies that there is an infinite number of such events $A_m$."
   ]
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   "source": [
    "**Exercise 2.10.4**.  Let $\\{ A_i : i \\in I \\}$ be a collection of events where $I$ is an arbitrary index set.  Show that\n",
    "\n",
    "$$ \\\n",
    "\\left( \\cup_{i \\in I} A_i \\right)^c = \\cap_{i \\in I} A_i^c \n",
    "\\quad \\text{and} \\quad\n",
    "\\left( \\cap_{i \\in I} A_i \\right)^c = \\cup_{i \\in I} A_i^c \n",
    "$$\n",
    "\n",
    "Hint: First prove this for $I = \\{1, \\dots, n\\}$."
   ]
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   "source": [
    "**Solution**.\n",
    "\n",
    "We can prove the result directly by noting that every outcome $\\omega$ belongs to or does not belong to both sides of each equality:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "& \\omega \\in \\left( \\cup_{i \\in I} A_i \\right)^c \\\\\n",
    "\\Longleftrightarrow& \\; \\text{not }\\left( \\omega \\in \\cup_{i \\in I} A_i  \\right) \\\\\n",
    "\\Longleftrightarrow& \\; \\forall i \\in I, \\text{not} \\left( \\omega \\in A_i \\right) \\\\\n",
    "\\Longleftrightarrow& \\; \\forall i \\in I, \\omega \\in A_i^c \\\\\n",
    "\\Longleftrightarrow& \\; \\omega \\in \\cap_{i \\in I} A_i^c\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "& \\omega \\in \\left( \\cap_{i \\in I} A_i \\right)^c  \\\\\n",
    "\\Longleftrightarrow&\\; \\text{not }\\left( \\omega \\in \\cap_{i \\in I} A_i  \\right) \\\\\n",
    "\\Longleftrightarrow&\\; \\text{not } \\left( \\forall i \\in I, \\omega \\in A_i \\right) \\\\\n",
    "\\Longleftrightarrow&\\; \\exists i \\in I, \\text{not } \\omega \\in A_i \\\\\n",
    "\\Longleftrightarrow&\\; \\exists i \\in I, \\omega \\in A_i^c \\\\\n",
    "\\Longleftrightarrow&\\; \\omega \\in \\cup_{i \\in I} A_i^c \n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.5**.  Suppose we toss a fair coin until we get exactly two heads.  Describe the sample space $S$.  What is the probability that exactly $k$ tosses are required?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  The sample space is a set of coin toss results sequences containing two heads, and ending in heads:\n",
    "\n",
    "$$ S = \\left\\{ (r_1, \\dots, r_k) : r_i \\in \\left\\{ \\text{head}, \\text{tails} \\right\\} , \n",
    "\\Big| \\left\\{ r_j = \\text{head} \\right\\} \\Big|= 2, r_k = \\text{head} \\right\\} $$\n",
    "\n",
    "The probability of requiring exactly $k$ tosses is 0 if $k < 2$, as there are no such sequences in the event space.\n",
    "\n",
    "The probability of stopping after $k$ tosses is the probability of obtaining exactly 1 head in the first $k - 1$ tosses, in a procedure that would not stop after any number of tosses, followed by the probability of getting a head in the $k$-th toss.  This value is\n",
    "\n",
    "$$ \\left((k-1) \\left(\\frac{1}{2}\\right)^{k - 1} \\right) \\left(\\frac{1}{2}\\right) = \\frac{k - 1}{2^k}$$\n",
    "\n",
    "Note that, besides this combinatorial argument, we can verify that these probabilities do indeed add up to 1:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\frac{1}{1 - x} &= \\sum_{k = 0}^\\infty x^k \\\\\n",
    "\\frac{d}{dx} \\frac{1}{1 - x} &= \\sum_{k = 0}^\\infty \\frac{d}{dx} x^k \\\\\n",
    "\\frac{1}{(1 - x)^2} &= \\sum_{k = 0}^\\infty k x^{k - 1} \\\\\n",
    "\\frac{x}{(1 - x)^2} &= \\sum_{k = 0}^\\infty k x^k \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "so, for $x = 1/2$, $\\sum_{k = 0}^\\infty 2^{-k} k = 2$, and so\n",
    "\n",
    "$$ \\sum_{k = 0}^\\infty \\frac{k}{2^{k + 1}} = 1 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.6**.  Let $\\Omega = \\{1, 2, \\dots\\}$.  Prove that there does not exist a uniform distribution on $\\Omega$, i.e. if $\\mathbb{P}(A)  =\\mathbb{P}(B)$ whenever $|A| = |B|$ then $\\mathbb{P}$ cannot satisfy the axioms of probability."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Assume that such a distribution exists, and let $\\mathbb{P}(\\{1\\}) = p$.  Since the distribution is uniform, the probability associated with any set of size 1 is $p$, and the probability associated with any set of size $n$ is $np$.  \n",
    "\n",
    "- If $p > 0$, then a finite set $A$ of size $|A| = \\lceil 2 / p \\rceil$ would have probability value $\\mathbb{P}(A) = \\lceil 2 / p \\rceil p \\geq (2 / p) p = 2$, which is greater than $1$ -- a contradiction.  \n",
    "\n",
    "- If $p = 0$, then any finite set $A$ must have $\\mathbb{P}(A) = 0$.  But then $\\mathbb{P}(\\Omega) = \\sum_i \\mathbb{P}(\\{ i \\}) = \\sum_i 0 = 0$, instead of $1$ -- a contradiction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.7**.  Let $A_1, A_2, \\dots$ be events.  Show that\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cup_{n=1}^\\infty A_n \\right) \\leq \\sum_{n=1}^\\infty \\mathbb{P}(A_n) $$\n",
    "\n",
    "Hint: Define $B_n = A_n - \\cup_{i=1}^{n-1} A_i$.  Then show that the $B_n$ are disjoint and that $\\cup_{n=1}^\\infty A_n  =\\cup_{n=1}^\\infty B_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Following the hint, let $B_n = A_n - \\cup_{i=1}^{n-1} A_i$.  \n",
    "\n",
    "- Note that, for $i < j$, $B_i$ and $B_j$ are disjoint, since all elements of $B_i$ must be elements of $A_i$, and all elements of $A_i$ are explicitly excluded on the definition of $B_j$.\n",
    "- Also note that $\\cup_{n=1}^\\infty A_n = \\cup_{n=1}^\\infty B_n$:  $A_n = \\cup_{i=1}^n B_i$ by construction, so $\\cup_{n=1}^\\infty A_n = \\cup_{n=1}^\\infty \\cup_{i=1}^n B_i = \\cup_{n=1}^\\infty B_n$, since $B_i \\cup B_i = B_i$ and we can include each $B_i$ only once in the expression.\n",
    "\n",
    "Now, we have:\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cup_{n=1}^\\infty A_n \\right) = \\mathbb{P}\\left( \\cup_{n=1}^\\infty B_n \\right) = \\sum_{n=1}^\\infty \\mathbb{P}(B_n) \\leq \\sum_{n=1}^\\infty \\mathbb{P}(A_n) $$\n",
    "\n",
    "since $B_n \\cup \\left(\\cup_{i=1}^{n-1} A_i\\right) = A_n$ and so $\\mathbb{P}(B_n) \\leq \\mathbb{P}(A_n)$ for every $n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.8**.  Suppose that $\\mathbb{P}(A_i) = 1$ for each $i$.  Prove that\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cap_{i=1}^\\infty A_i \\right) = 1 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Using the result from exercise 4,\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cap_{i=1}^\\infty A_i \\right) = 1 - \\mathbb{P}\\left(\\left( \\cap_{i=1}^\\infty A_i \\right)^c\\right)\n",
    "= 1 - \\mathbb{P}\\left( \\cup_{i=1}^\\infty A_i^c \\right)$$\n",
    "\n",
    "Using the result from exercise 7,\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cup_{i=1}^\\infty A_i^c \\right) \\leq \\sum_{i=1}^\\infty \\mathbb{P}(A_i^c) =  \\sum_{i=1}^\\infty \\left(1 - \\mathbb{P}\\left( A_i \\right) \\right) = \\sum_{i=1}^\\infty 0 = 0 $$\n",
    "\n",
    "so the equality holds, since a probability is non-negative.  Therefore,\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cap_{i=1}^\\infty A_i \\right) = 1 - \\mathbb{P}\\left( \\cup_{i=1}^\\infty A_i^c \\right) = 1 - 0 = 1 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.9**.  For fixed $B$ such that $\\mathbb{P}(B) > 0$, show that $\\mathbb{P}(\\cdot | B)$ satisfies the axioms of probability."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "- Axiom 1: $\\mathbb{P}(\\cdot | B) = \\frac{\\mathbb{P}(\\cdot B)}{\\mathbb{P}(B)} \\geq 0$, since $\\mathbb{P}(\\cdot B) > 0$.\n",
    "- Axiom 2: $\\mathbb{P}(\\Omega | B) =  \\frac{\\mathbb{P}(\\Omega B)}{\\mathbb{P}(B)} = \\frac{\\mathbb{P}(B)}{\\mathbb{P}(B)} =1$.\n",
    "- Axiom 3:  Assuming $A_1, A_2, \\dots$ are disjoint,\n",
    "  $$ \\mathbb{P} \\left( \\cup_{i=1}^\\infty A_i | B \\right) = \\frac{\\mathbb{P} \\left( B \\left( \\cup_{i=1}^\\infty A_i \\right) \\right)}{\\mathbb{P}(B)} \n",
    "=  \\frac{\\mathbb{P}\\left( \\cup_{i=1}^\\infty \\left( A_i B \\right) \\right)}{\\mathbb{P}(B)}\n",
    "= \\frac{\\sum_{i=1}^\\infty \\mathbb{P}(A_i B)}{\\mathbb{P}(B)} = \\sum_{i=1}^\\infty \\frac{\\mathbb{P}(A_i B)}{\\mathbb{P}(B)}\n",
    "= \\sum_{i=1}^\\infty \\mathbb{P}(A_i | B)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.10**.  You have probably heard it before.  Now you can solve it rigorously.  It is called the \"Monty Hall Problem\".  A prize is placed at random between one of three doors.  You pick a door.  To be concrete, let's suppose you always pick door 1.  Now Monty Hall chooses one of the other two doors, opens it and shows to you that it is empty.  He then gives you the opportunity to keep your door or switch to the other unopened door.  Should you stay or switch?  Intuition suggests it doesn't matter.  The correct answer is that you should switch.  Prove it.  It will help to specify the sample space and the relevant events carefully.  Thus write $\\Omega = \\{ (\\omega_1, \\omega_2) : \\omega_i \\in \\{ 1, 2, 3 \\} \\}$  where $\\omega_1$ is where the prize is and $\\omega_2$ is the door Monty opens."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Following the provided notation, the event space is\n",
    "\n",
    "$$ \\Omega = \\{ (1, 2), (1, 3), (2, 3), (3, 2) \\} $$\n",
    "\n",
    "$\\mathbb{P}[ \\omega_2]$ = probability of opening an empty door. The probability and the reward associated with switching for each outcome are:\n",
    "\n",
    "| $\\omega$ | $\\mathbb{P}$              | $R$    |\n",
    "|----------|---------------------------|--------|\n",
    "| $(1, 2)$ | $\\frac{1}{3}\\frac{1}{2}$  | $0$    |\n",
    "| $(1, 3)$ | $\\frac{1}{3}\\frac{1}{2}$  | $0$    |\n",
    "| $(2, 3)$ | $\\frac{1}{3} 1$             | $1$    |\n",
    "| $(3, 2)$ | $\\frac{1}{3} 1$             | $1$    |\n",
    "\n",
    "Therefore, \n",
    "\n",
    "$$\\mathbb{P}[ R | \\omega_2 = 2 ] = \\frac{\\mathbb{P}(\\{(3, 2)\\})}{\\mathbb{P}(\\{ (3, 2), (1, 2) \\})}= \\frac{\\frac{1}{3} 1}{\\frac{1}{3} 1 + \\frac{1}{3}\\frac{1}{2}} = \\frac{2}{3}$$ \n",
    "\n",
    "and, similarly, $\\mathbb{P}[ R | \\omega_3 = 3 ]$, and so $\\mathbb{P}[R] = \\frac{2}{3}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.11**.  Suppose that $A$ and $B$ are independent events.  Show that $A^c$ and $B^c$ are independent events."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\mathbb{P}(A^c B^c) &= \\mathbb{P}((A \\cup B)^c)  = 1 - \\mathbb{P}(A \\cup B) \\\\\n",
    "&= 1 - \\left( \\mathbb{P}(A) + \\mathbb{P}(B) - \\mathbb{P}(AB) \\right)  \\\\\n",
    "&= 1 - \\mathbb{P}(A) - \\mathbb{P}(B) + \\mathbb{P}(A) \\mathbb{P}(B) \\\\\n",
    "&= 1 - (1 - \\mathbb{P}(A^c)) - (1 - \\mathbb{P}(B^c)) + (1 - \\mathbb{P}(A^c)) (1 - \\mathbb{P}(B^c)) \\\\\n",
    "&= \\mathbb{P}(A^c) \\mathbb{P}(B^c)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.12**.  There are three cards.  The first card is green on both sides, the second is red on both sides, and the third is green on one side and red on the other.  We choose a card at random and we see one side (also chosen at random).  If the side we see is green, what is the probability that the other side is also green?  Many people intuitively answer 1/2.  Show that the correct answer is 2/3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  There are 6 potential card sides to be chosen, all with equal probability, of which only 3 are green -- one belongs to the red / green card, and two belong to the green / green card.  The probability that the other side is also green is the probability that the a side on the green / green card was chosen, which is 2 / 3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.13**.  Suppose a fair coin is tossed repeatedly until both a head and a tail have appeared at least once.\n",
    "\n",
    "**(a)** Describe the sample space $\\Omega$.\n",
    "\n",
    "**(b)** What is the probability that three tosses will be required?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**.  The sample space consists of the sequence of $k$ identical coin toss results and a coin toss result with the opposite value,\n",
    "\n",
    "$$ \\Omega = \\{ (r_1, \\dots, r_k, r_{k+1}) : r_i \\in \\{ \\text{head}, \\text{tails} \\}, r_1 = \\dots = r_k \\neq r_{k + 1} \\} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**  Exactly 3 tosses will be required if the first 3 results are $(h, h, t)$ or $(t, t, h)$.\n",
    "\n",
    "If we map all infinite coin toss sequences to $\\Omega$ by truncating it whenever the stop condition occurs, the probability of a (single-outcome) event in $\\Omega$ is the same as the probability of all outcomes mapped into it.  In particular, the probability of a sequence with its first 3 symbols being a specific sequence is 1/8, and so the probability of the desired outcome is 1/8 + 1/8 = 1/4."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.14**.  Show that if $\\mathbb{P}(A) = 0$ or $\\mathbb{P}(A) = 1$ then $A$ is independent of every other event.  Show that if $A$ is independent of itself then $\\mathbb{P}(A)$ is either 0 or 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  \n",
    "\n",
    "If $\\mathbb{P}(A) = 0$, then $\\mathbb{P}(AB) = \\mathbb{P}(A) - \\mathbb{P}(A - B) = 0 -\\mathbb{P}(A - B) \\leq 0$, and since probabilities are non-negative we must have $\\mathbb{P}(AB) = 0$.  Therefore $\\mathbb{P}(AB) = \\mathbb{P}(A) \\mathbb{P}(B) = 0$ for all events $B$, and $A$ is independent of every other event.\n",
    "\n",
    "If $\\mathbb{P}(A) = 1$, then $\\mathbb{P}(A^c) = 0$, and so $A^c$ and $B$ are independent for every other event $B$.  Then, from the result in exercise 10, $A$ is also independent from every other event $B^c$ -- which covers all potential events, since every event has a complement.\n",
    "\n",
    "If $A$ is independent of itself, $\\mathbb{P}(AA) = \\mathbb{P}(A) \\mathbb{P}(A)$, so $\\mathbb{P}(A) = \\mathbb{P}(A)^2$ or $\\mathbb{P}(A) ( \\mathbb{P}(A) - 1 ) = 0$.  Therefore $\\mathbb{P}(A) = 0$ or $\\mathbb{P}(A) = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.15**.  The probability that a child has blue eyes is 1/4.  Assume independence between children.  Consider a family with 5 children.\n",
    "\n",
    "**(a)** If it is known that at least one child has blue eyes, what is the probability that at least 3 children have blue eyes?\n",
    "\n",
    "**(b)** If it is known that the youngest child has blue eyes, what is the probability that at least 3 children have blue eyes?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)** Represent the sample space as\n",
    "\n",
    "$$ \\Omega = \\{ (x_1, x_2, x_3, x_4, x_5) : x_i \\in \\{ 0, 1 \\} \\} $$\n",
    "\n",
    "where $x_i = 1$ if the $i-$th child (youngest to oldest) has blue eyes.\n",
    "\n",
    "- \"At least one child has blue eyes\" is the event $A = \\Omega - \\{ (0, 0, 0, 0, 0) \\}$.  \n",
    "- \"At least 3 children have blue eyes\" is the event $B$ with 3 children with blue eyes, 4 children with blue eyes, or 5 children with blue eyes.\n",
    "- The intersection of these events is $BA = B$.\n",
    "\n",
    "Let $p = 1/4$ be the probability at a given child will have blue eyes.  The desired probability is then:\n",
    "\n",
    "$$\\mathbb{P}(B | A) = \\frac{\\mathbb{P}(BA)}{\\mathbb{P}(A)} = \\frac{\n",
    "\\binom{5}{3} p^3 (1 - p)^2 + \\binom{5}{4} p^4 (1 - p) + \\binom{5}{5} p^5\n",
    "}{1 - \\left(1 - p \\right)^5} = \\frac{106}{781} \\approx 0.1357 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)** \n",
    "\n",
    "- \"The youngest child has blue eyes\" is the event $C = \\{ \\omega = (1, x_2, x_3, x_4, x_5) : \\omega \\in \\Omega \\}$.\n",
    "- The intersection of events $B$ and $C$ is $BC$, the set of outcomes starting with 1 and having the other 4 dimensions having 2, 3, or 4 values 1; $BC = \\{ \\omega = (1, x_2, x_3, x_4, x_5) : \\omega \\in \\Omega, x_2 + x_3 + x_4 + x_5 \\geq 2 \\}$.\n",
    "\n",
    "The desired probability is then\n",
    "\n",
    "$$ \\mathbb{P}(B | C) = \\frac{\\mathbb{P}(BC)}{\\mathbb{P}(C)} = \\frac{p \\left(\\binom{4}{2} p^2 (1-p)^2 + \\binom{4}{3} p^3 (1 - p) + \\binom{4}{4} p^4 \\right)}{ p } = \\frac{67}{256} \\approx 0.2617 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.16**.  Show that\n",
    "\n",
    "$$  \\mathbb{P}(ABC) = \\mathbb{P}(A | BC) \\mathbb{P}(B | C) \\mathbb{P}(C) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "$$ \\mathbb{P}(A | BC) \\mathbb{P}(B | C) \\mathbb{P}(C) = \\frac{\\mathbb{P}(ABC)}{\\mathbb{P}(BC)}  \\frac{\\mathbb{P}(BC)}{\\mathbb{P}(C)} \\mathbb{P}(C) = \\mathbb{P}(ABC) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.17**.  Suppose $k$ events for a partition of the sample space $\\Omega$, i.e. they are disjoint and $\\cup_{i=1}^k A_i = \\Omega$.  Assume that $\\mathbb{P}(B > 0)$.  Prove that if $\\mathbb{P}(A_1 | B) < \\mathbb{P}(A_1)$ then $\\mathbb{P}(A_i | B) > \\mathbb{P}(A_i)$ for some $i = 2, \\dots, k$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "We have\n",
    "\n",
    "$$ B = B\\Omega = B \\left( \\cup_{i=1}^k A_i \\right) = \\cup_{i=1}^k A_i B $$\n",
    "\n",
    "and so\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\cup_{i=1}^k A_i B \\right) = \\mathbb{P}(B) \n",
    "\\Longleftrightarrow \\sum_{i=1}^k \\frac{\\mathbb{P}(A_i B)}{\\mathbb{P}(B)} = 1\n",
    "\\Longleftrightarrow \\sum_{i=1}^k \\mathbb{P}(A_i | B) = 1\n",
    "\\Longleftrightarrow \\sum_{i=1}^k \\mathbb{P}(A_i | B) = \\sum_{i=1}^k \\mathbb{P}(A_i)\n",
    "$$\n",
    "\n",
    "If we assume that $\\mathbb{P}(A_i | B) \\leq \\mathbb{P}(A_i)$ for all $i$ and $\\mathbb{P}(A_1 | B) < \\mathbb{P}(A_1)$, then we must have $\\sum_{i=1}^k \\mathbb{P}(A_i | B) < \\sum_{i=1}^k \\mathbb{P}(A_i)$, a contradiction.  Therefore the desired statement must hold."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.18**.  Suppose that 30% of computer users use a Macintosh, 50% use Windows and 20% use Linux.   Suppose that 65% of the Mac users have succumbed to a computer virus, 82% of the Windows users get the virus and 50% of the Linux users get the virus.  We select a person at random and learn that her system was infected with the virus.  What is the probability that she is a Windows user?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  The event space can be described as:\n",
    "\n",
    "|  outcome          |  probability  |\n",
    "|-------------------|---------------|\n",
    "| Mac, no virus     | 30% * 35%     |\n",
    "| Mac, virus        | 30% * 65%     |\n",
    "| Windows, no virus | 50% * 18%     |\n",
    "| Windows, virus    | 50% * 82%     |\n",
    "| Linux, no virus   | 20% * 50%     |\n",
    "| Linux, virus      | 20% * 50%     |\n",
    "\n",
    "The desired conditional probability is\n",
    "\n",
    "$$ \\mathbb{P}(\\text{Windows} | \\text{virus}) = \\frac{\\mathbb{P}(\\text{Windows}, \\text{virus})}{\\mathbb{P}(\\text{virus})}\n",
    "= \\frac{0.50 \\cdot 0.82}{0.30 \\cdot 0.65 + 0.50 \\cdot 0.82 + 0.20 \\cdot 0.50} \\approx 0.5816\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.19**.  A box contains 5 coins and each has a different probability of showing heads.  Let $p_1, \\dots, p_5$ denote the probability of heads on each coin.  Suppose that\n",
    "\n",
    "$$ p_1 = 0, \\quad p_2 = 1/4, \\quad p_3 = 1/2, \\quad p_4 = 3/4, \\quad \\text{and } p_5 = 1$$\n",
    "\n",
    "Let $H$ denote \"heads is obtained\" and let $C_i$ denote the event that coin $i$ is selected.\n",
    "\n",
    "**(a)**  Select a coin at random and toss it.  Suppose a head is obtained.  What is the posterior probability that coin $i$ was selected ($i = 1, \\dots, 5$)?  In other words, find $\\mathbb{P}(C_i | H)$ for $i = 1, \\dots, 5$.\n",
    "\n",
    "**(b)**  Toss the coin again.  What is the probability of another head? In other words find $\\mathbb{P}(H_2 | H_1)$ where $H_j$ means \"heads on toss $j$\".\n",
    "\n",
    "**(c)**  Find $\\mathbb{P}(C_i | B_4)$ where $B_4$ means \"first head is obtained on toss 4\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**  We have:\n",
    "\n",
    "$$ \\mathbb{P}(C_i | H) = \\frac{\\mathbb{P}(C_i H)}{\\mathbb{P}(H)} = \\frac{\\mathbb{P}(C_i H)}{\\sum_j \\mathbb{P}(C_j H)} = \\frac{\\mathbb{P}(C_i)\\mathbb{P}(H | C_i)}{ \\sum_j \\mathbb{P}(C_j)\\mathbb{P}(H | C_j)}$$\n",
    "\n",
    "Assuming that the coin selection is uniformly random, $\\mathbb{P}(C_i) = 1/5$ for $i = 1, \\dots, 5$, and the above simplifies to\n",
    "\n",
    "$$ \\frac{\\mathbb{P}(H | C_i)}{ \\sum_j \\mathbb{P}(H | C_j)} = \\frac{p_i}{\\sum_j p_j}$$\n",
    "\n",
    "Therefore,\n",
    "\n",
    "$$ \\mathbb{P}(C_1 | H) = 0\n",
    "\\quad\n",
    "\\mathbb{P}(C_2 | H) = 1/10\n",
    "\\quad\n",
    "\\mathbb{P}(C_3 | H) = 1/5\n",
    "\\quad\n",
    "\\mathbb{P}(C_4 | H) = 3/10\n",
    "\\quad\n",
    "\\mathbb{P}(C_5 | H) = 2/5\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)** We have:\n",
    "\n",
    "$$ \\mathbb{P}(H_2 | H_1) = \\frac{\\mathbb{P}(H_2 H_1) }{\\mathbb{P}(H_1) } = \\frac{\\sum_j \\mathbb{P}(C_j H_1 H_2)}{\\sum_j \\mathbb{P}(C_j H_1)} = \\frac{\\sum_j (1/5) p_j^2}{\\sum_j (1/5) p_j} = \\frac{3}{16} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(c)** We have:\n",
    "\n",
    "$$ \\mathbb{P}(C_i | B_4) = \\frac{\\mathbb{P}(C_i B_4)}{\\mathbb{P}(B_4)}= \\frac{\\mathbb{P}(C_i B_4)}{\\sum_j \\mathbb{P}(C_j B_4)}$$\n",
    "\n",
    "But $\\mathbb{P}(C_i B_4) = (1/5) (1 - p_i)^3 p_i$ -- selecting coin $i$, then obtaining 3 tails followed by a head on that coin -- so\n",
    "\n",
    "$$\n",
    "\\mathbb{P}(C_1 | B_4) = 0\n",
    "\\quad\n",
    "\\mathbb{P}(C_2 | B_4) = \\frac{27}{46}\n",
    "\\quad\n",
    "\\mathbb{P}(C_3 | B_4) = \\frac{8}{23}\n",
    "\\quad\n",
    "\\mathbb{P}(C_4 | B_4) = \\frac{3}{46}\n",
    "\\quad\n",
    "\\mathbb{P}(C_5 | B_4) = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.20 (Computer Experiment)**.  Suppose a coin has probability $p$ of falling heads.  If we flip the coin many times, we would expect the proportion of heads to be near $p$.  We will make this formal later.  Take $p = .3$  and $n = 1000$ and simulate $n$ coin flips.  Plot the proportion of heads as a function of $n$.  Repeat for $p = .03$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "np.random.seed(0)\n",
    "\n",
    "n = 1000\n",
    "X1 = np.where(np.random.uniform(low=0, high=1, size=n) < 0.3, 1, 0) \n",
    "X2 = np.where(np.random.uniform(low=0, high=1, size=n) < 0.03, 1, 0) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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IERERETuzbxBWa4SIiIiIrdk3CKs1QkRERMTW7BeEtVlORERERLBjEG6girCIiIiIrdk4CGuznIiIiIid2TAIqxIsIiIiIrYMwvXUGiEiIiJia/YLwtosJyIiIiLYMQg3UEVYRERExNb8PL0Az7FPEJ69dA+fbsjk4jM74XBa3Dium6eXJCIiIuJxNgzC9QHYRlMjUtML+D6jmL0FFfj5GG4Y2xVjjKeXJSIiIuJRao04zS1Ly+OrTQcBKKqoJa+shl255VTXOTy8MhERERHPsl8QttlmuatfWdXksVmLdtH/T1+zK7fMAysSERER8Q72C8INbFARto7yNX68PpMah5PvtuW08YpEREREvId9g/BpXhF2Oi3OfPSbJo8nRQdT53R97St357f1skRERES8hg2DsD02y+WVVVNUUQtA7w5hfHPXeGaO6cp1o5Ld16zaXUCd4/T+PoiIiIgcjQ2nRtQ7zVsjFu3IdX/810v6kxIfxsM/68v2g6UA+PsaSqvr2JRVQnCALz3jwzy1VBERERGPsF9F2Dr9K8LL0vK494ON7s87R7Vzf5wSF0pCRBDTBnYC4Jn5Ozn3qcUsT8tr83XKqeu77TnsyC719DJEREROiH0rwqdxj3BmYaX74y9/PY64sCD35z4+hi9+PY7gAF9+yCxifv2GuQXbchjdI6bN1yqnlqpaB3e9t4GvNh0kKiSAT+4YQ+eoYE8vyytZlqV53SIiXs5+FeEGztOnIpyWU4rDeSjYF1XWuD/uk9C05SEqJIAgf19Gdz8UfBfvzG1yndiDZVmNJozUOZw8/MkmHv9qK5ZlUVFTh9NpsXRnHr0fmuueS11d6+CmN1KpqKljd26Z+zVKq2pJyykjo7CCjMIKj3xNzamqPfrs7OySKh74cCPPLth51GkrPyazqJLv9xexKbOYtBzX9+LmN9dy1csrySqq5MN1GZrdLSLihWxYET69WiNyS6uZ8u8lPHrRGVw1ogsAB4qrAJjSL/5HK1Kjukfz+vJ0AHZkl3GwuIoOEUFHvV5OP8WVtfxz3na2Z5fyxszh7Mwu42fPLnU/vyunnMU7chnWNZJlaYemjMyaMZh2AX788rXVXPzcMnZkl3HHWd25d0pv7vtgI99sySYyJIDqWgcf3j6GHnGhnvjy3P67ci9/+XwLz101mMl94xs9t/VACef/Z4n78+LKWpJjQggL8mfawI6AawpLjcNJkL9vo1+bU1rFcwvSmLNiLz4GAvx88DGG60cn882WbAAm/nMhNXVOPlibwaxrhhAe5H+Sv1oREWkpGwbheqdJEM4sqsThtFi9J98dhA8WV9E9NoQXrxn6o792ZNdojIEO4UEcKK5iyc5cLh/auS2WLV6gps7JmX/5moY3Ey57YTmbs0rczw9LjuTbra4w1xCCbxzblVsmdCc2LBCA+8/vzd++3AbAc9/tIjzIn6+3ZGNZFrml1fj7Gn75+mo+un0MMaGBbfjVHVJd5+DZBWnU1Dm5/a213DahO/O35fD3ywbg7+vDI59tBmBIUiR9EsJ4ecke96/NLa3mhrFdeeSzzXy8IYvXfjmMwV0iAdh+sJQp/17svrZrTAh78ytIim7H8wt30SUqmEcvPoM73lrHwMQIVu8p4OezVjBn5nDiww/9wFlV6+Dj9Zm8vjydWyZ045IzE9voOyMiIvYLwu7NcqfH25S760+H27C/yP3YgeIqEiLaHe2XuEUE+3PFsC707xTBU9/uYPHOPAXh05hlWSzcnsuIblEEB/ixaEeuOwRPHZDA5xsPADAuJYZnrxqMZVnc+t+1jO8Zy+IduTx4YV/6dQxv9C7DTeO6UeuwSI4O4Y0V6Tz+1TaMgbduHMEPGcUMSYpkxuxV3DgnlXdvHtmkotoWPl6fycGSKp696kxeWrybpxekAXDVyyspqaoD4FeTevC7c3thWRYhAX589n0WfRLCefTzLewvqOC/q/YBMOOVVbxy7VD8/Xx4+BNXgB6eHMVjl5xB99hQ8sqqCfL35bHPt3DJmZ0Y3SOG1X88myA/X5btyuPWN9dy6fPLmTNzGD3iXG1Lj3y2hXdWu17/rve+Z39BJdeOSmJXbjlDkiLb+tslImIr5nj64VrD0KFDrdTU1La/8ZuXwq750O8SuPz1tr//CXruuzQGdW7P419tZVKvOPc/6gDrH5pMZEgAI/82n7EpMfzz8oEtft2739/Ad9tyWPvgZHx8tMHndFNQXsPv3t/Ad9tzubB/As9edSa3v7WONekFLP39JPx9fXjw402c2aU9lw9JPK5NXoXlNVz2wnJ6dQjjhRlD3I/P3XSQ295ay3n9OvDcVYPb9M+Xw2lxzpOLCAn05bM7x1JSWccfPvqBkd2ieHHxbjIKKxnRNYqXrhlKRPChlgXLsnA4LX73v+/5ZEMWgX4+/L/bRvO7979nT345NXWud5TuOqcnvzknpcXr2ZRZzPWvraHW4eSV64YSGRzAuU8twmnBs1edyYJtOXy4LpOIdv4UV9Zy51k9uHlCN7YfLCU+LIj88mrO7NK64diyLJam5TEgsT0R7dS2ISKnJ2PMWsuymrxVbr+KcINTsDWius7BE/O20z02hF255WzKLGn0/IaMIsb1iCG3rJqEY+z1HZ8Sy4frMtmUVcyAxPatuWzxAk/M2853210bIr/44QC9FoQxf2sOM0Ymuau0j1/a/4TuERkSwNd3jW8yj+W8Mzrwxwv68NgXW/m/udv4wwV9Tug+x+KrTQfYk1fO81cPxhhDRLA/z109GIDz+yeQUVjJoM5N/7wbY/DzNTz580F0iQomIaIdZ3SK4N2bR3L9a6v5PqOYXwztzI3juh7Tes7oFMFHt4/muldXc/Urq+geG+qqzt87kejQQC7sn0DnyGD+M38n/TqG8+x3afx31V6KKmoJCfClotbBvVN6cduE7k1+WDneKRVf/HCAO99eT1J0MC9cPYSdOaUMSYokMVLTQETk9GfDIFz/z7Tz1GuNyC9zTYPYlVve5DljYP3eQl5blo7DaR3zprexKa4JEkt25ikIn2YcTov59b2+/7x8IF/+cIAnv9kBwKWDO7Xqvfx8mx9Ec8PYruwrqOClxbvpHBXMNSOTANeEilqHRbuA1m+ZsCyL57/bRbeYEKb069Dk+ZjQwJ/sW/b1Mfzu3F7uzyNDAnj/1lEUltce98bSzlHBfHDbaG6Ys4b1+4q4d0ovouvXYYzhrsk9uWpEF+LCApm1aDd/n7uNMzqFsyWrhGFJUfxj7nbSssv426X9CfL3ZV9+Bf+ev4OF23P508/6ctGg5n9P88uq8ffzITzIH8uyWLu3kOSYEP7y2RZ6xIVSWlXLtGeXUue0iGjnz1O/GMievAqGJEU2+8OCiMjpwIZBuN4peLJcXln1UZ/rGRfGl5sOkpbj6hnuEH5s/0jHhAbSr2M4i3fkcsdZPU5oneJdVu7OJ6e0mueuGsyFAxI4u3ccU59ZSkQ7f/p1DG+TNRhjeHhqXzIKK/nTJ5voGBHE2X3i+eNHm/hmazbv3zLS3TPbWhbtyGXLgRL+cdkAfFuxHSPQz5cOEScW3KNCAnj7xpF8uzW72ZDesJnutonduWxwJ2LDAimrriM00I9nFqTx5Dc7SM8v54UZQ7jutdXsyXP9cPybdzewJauEUd2jyS2tZvqQRBxOiwc/3sS7a/YTGxbIryb1cPc3hwb6UV5Tx4vXDCExMpi7399ATZ2Tkqo6Zr7ual3z8zE8cEEffjk6mT355XSPDWVLVgl/n7uNX5/dgyFJUSf0vQBYt6+QQD8f+nWMOOHXEhE5FvYLwqfwyXLNBeEO4UF8fMcYnvpmB++l7j/0+HFUq8alxPLKkt3uf3CldVXXOcgurqZLtOst5+VpeQQH+p3UaltxRS3/S91PaKAfZ/eJA1xVza9+O446R9se+ODn68MzV57JlS+v5I631/GfK87kw/UZ1DosZryymg9uG9Wqb8c/v3AXCRFBXHxm61a9W0u7AF9+Vj+e7cfE1YfisPqxa78+O4WUuFDufv97Jv1zIeU1Dib1juMPF/RmzvK9vLh4Ny8u3g243uHp1SGMd9e4/m4I8D20ya9nfCgHi6u4blSyu+/4zRtGAK5JFo99sYXCilpq65w8+vkW3liRzt78Cq4Y1pkfMovZnFXC0rQ8rhzemcU78rhtYneuGNb5mP9M7ckr56qXV1LrsLhjYnd+dXYK/kd5Z0FEpLXZN+2cglMj8kprmjzWJyGMDhFBDOrSvlEQbsnUiCON7xnDrEW7WJ6Wx7nNVKnkxLy0aDf/nr+TT+4Yw8HiKm56M5X27fz55u4JJ2W02JasEi6ftZzyGgeXnNmp0cQGT82yDQn049Xrh3HZC8u55c21gGsm8X0fbOTa2at5/9ZRJ/y9OFBcydWvrGJ3bjkPT+1LgN/pF6rO759A56hgbn4jlUFd2jP7uqEYY3j04jPokxDO68v3MC4llleX7eHT72Fktyj+dkl/okMDeeqbHZzbN57BSZEY4wrHRwry9+Wxi10945ZlMXvpHv7vq22kxIW6Q/U/LhvAwh05/HflPnx9DA98+ANLduby+CUDGm08/DFOp8V9H3yPv68PU/q5Nv9+tz2Xp34xyOOzp0XEHmwchE+9inBufUU4JMCXmLBAMgor3ccnH15V/PcvBhEVEnDMrz80KYqwQD8WbMtRED4Ob65IZ1T36KO+xf/1lmwcTos7315Her7rxLXCilr+9Mlm9wau1pJZVMkFTx86JGLGyC6t+vonIiY0kDdmDueyF1YwuEt7zjsjgZjQQGbMXsX1r63mnZtGuqufR+N0Wvzl8y306hDGV5sOMnNMMhN7uSrezyxIY3d9H/0Vw0/fcYBndIpg0X1nYVk0qsJeNaKLe6b42JQYXly0i8cvHUDXmBAA/jyt3zHdxxjDjeO68bOBHYkOCWDl7gJ2ZJdy+dBELh+ayNxNB+kWG8p323P457ztfL9/CU9fOYghSVG8s3of8zYf5KGpfVm3t5BxKbFEhwZgWVBZ4+CdNftYk17IE9MHcPnQzpx/Rgce+PAHLnx6CQ+c35trRyU3mTKSXVLFv77ezlUjktx/7y3dmUdSdLCtjvsurqxlzZ4CJvaKPWpvvjRVXFFLWJCfpiOJmw2D8Km7WS6vrJrQQD8m9ool0M+X2ydGunvqesaHERzgS53D4qJBP/12a3MC/HwY3zOW+dtycDot/UXRAvsLKujYvh0Hiit56JPN9O4Qxme/Gtvkrd2ckip+yCxmWHIka9IL3Y/fMqEbLy7azdjV+/hoXSb3ndeLockt77l84MMfKK6s4dkrG48l+/z7LAAC/XxY/YdzWlyhaytJ0SEsvm8iPvUBbmhyFC/MGMJNc1K5cU4qc2YObzJzOLOokt+8s54L+ieQHBPsPhURXH3Qz155Jp9+n8XnGw8woWcsD/+sL8EBp/dfcT/VQnBWrzjOqv8B4UQ19C2PTYlxb64FV3UaoFeHMEZ2i+bX76zn5y+u5PrRyby5ci81dU6W7FyMo34TXmJkO3JKq3E4LQrKazirVyzTh7gOETnvjAQGJ0Xy+w828ufPtjB/Ww5XDu9CanohCRFBZBVXsjO7jKVpeXywNoNbJ3RnSFIkN8xJJcjfh3vO7cUvx3Rt1Z5wb7Fqdz5PfbuDgvIa7jirB2+t3Mfq9AJ6xYfx0NS+jE2JweG0julrL6qoYfbSPXy3PYcL+icwc0xXj8z6rq5zsCunnH0F5Uzu2+G4fv+yS6rwMYa8smoWbMvBsiyuGZlMRLA/B4orWbErn9BAP+58Zz2dI9txx1k9mDawo36IEBvOEX7jIti9ELpOgOs+bfv7n4Bfv7OejRlFfHfPxGb78K54aQWZRZUsuW/Scd/jw3UZ3P3+93x65xhNjzjM2r0FfLs1h/um9HJ/7/PLqhn1+AKuGtGFnvFh/OGjHwB44Pze3DKhe6Nf//6a/dz3/zby5a/H8fzCNPbmV/DZr8ZS63ByyfPL3KPwEiPbMfe34xv1aGcVVbJgWw5Xj+jS6Pe9uLKWYY99S43Dyf3n9+bWw+459Zkl+Pr48PHto9u0D/hEfbIhk9++t4Gze8fzwozBbM4qISkqmH99s53/rtznvi4uLBBfH1Sc7sAAACAASURBVEOvDmG0b+fP9uwyth44NE5wwe8m0C1Wb617QmlVLX/8aBOffp9FeJAfb94wgqe+3UFwgC8ZhZVszCgmLiyQnNJqzugUzuzrhjU6aQ9c7RjvrN7Po59vobK2adHi9+f1Zk9eGe+nZgDQPTaEpOgQFmzL4cwu7Xli+oBW33zpSdsOlnDev5c0efymcV2Zu/kg+wsqiQ8PJLukmgsHJHD/eb1/tDrudFosScvjiXnbGo3hTIgI4q7JPblscKI7jB4oruSpb3YwdUBHxqXEtMrfJ+v3FTJ300ESIoI4q3ccV728isyiSsD1e/mbc3pyYf+EZgPx0p15hAX5MbD+3YCDxVXc9EYqP2QWN7k2NNCPGSOT+OKHLPYXuF6/Z3woPsaw7WApnaPaceuE7kwfkkhNnZN2/r74+frgdFrszCkjJS60UYHh+/qDqwYkRpBXVuM+YVNODUebI2y/IDxnGuxZBMnj4PrP2/7+J+DqV1ZSWePgw9vHNPv8DxnFFFbUML5n7HHfo6C8hqGPfcOdk1K4e3LPZq95e9U+zukb527LsINb31zL3M0H+cdlA/jihwNM7BXLI59tAcDHuCrypVV19EkIZ1laHt/cPb7Rxq9b3kxlY0Yxy++fhNNy/UPfUInYeqCEac8upZ2/L2XVdfx8aGf+eGEf9ta3T9z+1jr2FVTwl4v6ce2oZMA1duyd1ft46JPN9OsYzvaDpbx/6ygGd4lkd24Zk/61iAcv7MON47q17TeqFbyxIt29oQsgOMCXihpXGJrSLx6AeZuz+cMFvbl5vCv8F5TXcM3sVTicFn+7tL/7GGTxDMuymLc5m+jQAIYd9g5HrcPJtgOl9OwQSkF5zU/uZUjPK+f5hWlM6BnHxowixqbEkJZTxvWjkzHG8N22HF5avJsHp/ahb0I4n2zI4s+fbaai2sFvzknh5vHdGlXN0/PK2VdQwbj6ivabK/fSLSa0UYX7x+SWVmOMq73Hsqz64B15XK1oLVVc6RprV1Hj4JFp/TirVxyfb8zC4bS4YngXqmodvL48nWcXpFFWXefu+b5+TDJ3nNWj2UNSnvx6u/swptsnuoJgTmk1j3+1je/3F9ErPgxjICzIj9KqOrYdLAVgVLdo7juv11EPdUnLKSMk0LfJ72tpVS2bMkt4cfEuFtbPM29gjGsP+9geMYzvGcMHazPYke0Kob85J4ULzkhwh9FPNmTym3c3AK6+96KKWvfaOrVvR68OYRRW1DB1QEdGdI3ihYW7+HLTAUIC/Hh4al+2Z5dy28TuRAUHMH9bDs9+l8b3+4sICfClvMZB56h2dIxox6o9BQAkRwdTUF5DckwI5/SJ55kFO6l1WO41j0uJ4bYJ3RnVPfqUKjjYlYJwg4YgnDQGfvll29//BJz378V0jgrm5Wub/D62qukvLKeqzsHnvxrX5Ln9BRWM+8d3XD86+Zh7Db1JncOJw7II9PvptwEdTovBj35DcWUt/r6GWkfj/2aiQwLIL6/hyuGduXNSCuf8axFjekQzsls0JVV13HFWdwb/5RumDep01EMrvth4gCB/H1anF/Diot0M7NyejRlFjab8Bfr58NmvxtIlKpjHv9zKnBV78fMxrH1wMhc+swTLgj9e2Ifb31oHwMoHzj7uWbee9p9vd/LUtzsaPTZn5nAm9Iylps7J/K3ZnN0nvtFGOKfTwmFZmjhgc7ml1fz508188cMB+iaE88TlA+jXMYLSqlrO/88SMgorObt3HEOSI/nH3O0AXDyoIw9O7dvsRk2n02LhjhxeW5bOkp15BAf48ttzUvAxhse+2EpEO3/uO68XVw7r0urtZE6nxY1vpLJkZy7v3jzyR0fVlVbV4utjKKms419fb+eDdRm0b+fPb85O4YIBCXy/v5iEiCBeWLSLLzYeoGd8KPec26vRfhDLsvjyh4P8Y9429w/iPgZmzRhCVlElzyxII7+8hnP7xnPvlF6kxLuq7nvyypnxiquq6+tjmDGiC1W1TqJCA7hoUEdueD3VXfEFV0X12lHJdIwI4o0Ve7ludDKjuke7v+YvNx3gP9/uZGdOGb3iwzi3XzwLt+fyQ2YxI7pGMaFXLC8v3k1hRS2R9YfkjO7e/A8zu3PLMMa4e+QP13Cq4rML0tie7Tq9cXu2K1iP7BZFQXkNO7LLaB/sT1FFLf07RTBtYEfeWrWX8Hb+ZBVVkVdWTf9OEdwyoRvn9evArtzyJpVk8Q4Kwg0agnCXUTBzbtvf/wQM++u3nNMnjscvHXBS7/PCwl38fe62ZoPU4h25XPvqamLDAln5wNle34tnWRbL0vIZ2S2qUS/YnW+vY39hJR/dNvon/8Jau7eAy15YwTUjk3hz5V58DDgtCPL34dXrhpFVXMU9//uel68dyuS+8by0eBd/+3KbOzTfOqE7sxbtcj//Y6rrHEx7Zpn7L+MGT195Jn/5bDMxoYH4+/q43wa8ZXw3HrigDxv2FzH9heXUOV3/PXdq345l9x9/i4ynWZbFh+sy6dcpnIyCSkZ1jyZEI/3kGMzddIAHP95MUUUNt07oTmZRJZ9syOT60V15e/VeqmqdjOwWxfCu0bywMI2QQD/um9Kb9fsKGZsSQ7+O4WQVuXr7n5i33f2641JiWLIzD3BVMWsdTlbtKWBgYgSPXnxGq7aU/fvbHfz72508elE/rql/N6ilNmcV87cvt7IsLb/Jc8OTo/jvjSOOOlGlps7JlgMl9O4QRm5ptbvNory6jleX7uGlxbspr6nj0sGJXD862b0BuGtMCClxoXy9JbvJa07pF881I5MZkhTZogN0HE6Lzzdm8Z/5O92bX/t3iuCtm0YQHuRPeXUdB4orW7UFxrIs9uSVkxwdgo+Pwem0qKpzEOjny6rd+fRPjGi0kbeq1sFH6zN5afFu9yxvgG6xIdw0rhuXnNmJkspaIoL9W1R0kZNLQbjBnJ/BnsXQeQTc8HXb3/84OZ0WKQ9+xa0TunHvlN4n9V47sks596nF/PWSM7h6RFKj5+YsT+dPn7retn77phFH/SncW6xJL+DyWSsa9dA6nBaDHvma0uo6/nX5QC6r36jTnAXbst0HC6z+w9m8sGgXVbUOOka0o09COOf0jceyLDZnldCvYzjGGGodTn72zFK2HSwlwNeHGoeTAD8fNjw8uUWbtzZlFjN91nJmjEhiaVoes68fRqf27fhuWw6/fH2N+7q/XdKf6UMS3f+YNQTwnvGhPHPlYHp1OH16JEWOR1FFDX/5fAsfrssE4NeTenD3ub3YX1DBW6v2MXNMMnHhQezMLuWBD38gde+hjayHv/sTHx7In3/Wj3P7dcDHwDdbspm76SAPTu1LZLA/n2zI4rEvtpJfXs2Vw7uw42Ap/RMj+O05PSmuqCU6NICQQD8qaxx8uzWbc/rEu8Pg/oIK4sIDCfTzpbSqlhW78jmrdxyLd+Ryw5xULhucyD8vH3Bcb71blsV3213tI0lRIRRW1DChVyw/G9jxhEYoFpTX8MLCNOascG2GDPDz4Z2bRjIkydUysSevnC1ZJfSMD+XVZen8bGDCcf9b4XC6WlD6dgwnITzIKyutDqfFN1sO8tqydAL8fCisqGFTZgl+PoY6p0VsWCDXjUri6hFJRLawjeZAcSWp6YXUOpxc0D/huDcxVtU6eG1ZOgM7RzCqW+MWjlqHk8fq+/BnjExiQGJ7quscrN9XRPtgf3p3aJsDl9qKgnCDhiDcaSjcNL/t73+ciipqGPSXb3h4al9mju16Uu9lWRbjn/iOnnFhzL5+WKPn/vzpZt6vn1d80Y+81e8tnvsujSfmbad9sD+L7zuL8CB/NmcVc+HTSwn08yE6JIAF90w86l8yd7y1ji9+OABA+v9d2OL7/pBRzD+/3s6MkUnc9EYq43vG8sbM4S3+9VW1DgL9fJr84/fnTzfz5sq9vH3jCIZ3jWr0vNNp8eH6TM7uHdfiv2xF7GDh9hxW7yngrsk9j9o60/Dfj68PlFbV8d6a/YzvGUtaThn/+vnAnwyOJVW1PPXNDuYsT8fHGByWRXRIAMWVtbQPDmBkt2g+q5/m0jEiiBvGdePNFemk51eQFB3MA+f34e3V+1i8I5cecaFkl1SRFB3MB7eO9sgkh5bIKqpk9tI9jE2JabXpJKcDy7JYsTuflxfvJqOwkuhQ19jBIH8fpg9JZOaYrkfdzLt2bwF3vfc9+woq3I9FhwRw9YguzBiZ5D5g56fuv3J3Ad1iQ7jrvQ0s3+V6V6BnfCjXjkrmokEdSd1byMuLd7N8Vz5B/j5U1TrpEB7EwZIq9+sMS45kTI8Y9uVXEBrkR3ZJFRcO6Mj+ggrmb80mvJ0/Vw7vwtm9406J6RsKwg0agnDHwXDzd21//+OUllPGOU8u4j9XDOKiQSf/pKw/f7qZd1bvY9n9k5i76SCBfj7MXrqHyOAASqtr6RYTypKduaz+4zle15NZ63C613TD62tYu6+Qoopafn12Cr85O4VZi3bxxLzt/PPygdzzv+/5/Xm9uW2iq1pcVeugqtbBrtxyAv18uOPtddTUOXnluqHHffzrJxsy6REX2irHxzqcFhmFFSRFN+13ExHP25FdSnl1Hf6+Pvzp080UltcQ1s7fPXHglvHdWLwzzz3lpFd8GE7LNaUA4LpRSSzakUtpVR0f3zHGVrORT2fbD5Yye+luPl6fRa3Tydm947lxXFf6dQznrve+p6Kmjl25ZWSXuM4L6JsQzrn94hmY2J7/rtzLgu05+PkYLuyfwC/HdHVPzQBX8N2eXcq8TdkUVtRQWeNwH7DlY+DxS/vjYwxzVqQ3mhIC8OjFZ3DRoI58uDaDt1bto7LWwfiesSRGtuP9NfvdM++bE+DnQ02dk/jwQH4xrAtXDu/s3ii5v6CClbvzyS+vce2v8TFkFFWSEhfGVcO7eGSkp4Jwg9enQvoSSBgEtyxq+/sfp9V7Cvj5iyv47w0jWrzD+UQs2ZnLNbNXc27feL7eku0ezWMMTB3QkWkDO3LTG6m8/sth7oMMvMF7a/bxf19t4+u7JhAdEsDgx75hSt8OlFXXsXB7DlMHdOS91P2EBPiy+S/nccPra1i9p4BF951FVEgAD3y4kQXbcjAYKmsdFFfWtkkVXkROT5ZlYVmwfn8RAxMj8PP1cfe/9u4QTkpcKE7L4p01+6moruPm8d1wOC2q6pw66v40lFtazZsr0nlz5V4KK2oJC/SjotZBSIAvJVV1jO4ezVO/GNRkpGB6XjmvL0/ng7UZlFXXMbhLe87uE8/avYUs2ZnbZBN3wyFKE3vGcU793hTLsli3r4i3V+0jwM8wbWAn9ybF5jidFqv2FNA5qh3ZJVX06hDO6j35VNU6GZIUSXRIAN9tz+WtVXtZtCMXA5zZJZI9eeUUlDc9CbeBpzZyHy0I2/e/slPsiOX8+lPlokPb5i3vEV2jCQvy45utrk0PDT+lWhZ0jQlhfM8YwoL8+HzjgTYLwrml1eSVVdMn4eh9S4t25FJYUcusRbu4cngXiipqGZIUyeCkSL7adMD9U3LXWFdF9f7zezPl34t5ZsFOV5/b6v1NXnN0j6P/RSEi8mOMMRiDu38WwNfHNHpnzwfDNSMP7cfw8zWEetk7bdI6YsMCufvcXtx+Vg8+XJfJR+szuGFsNyb0jKXO6TzqqZrJMSH8eVo/fnduT/6XmsGcFenuTZxB/j5M6BnHWb1jObt3POn55Yw4onUOXH8WhyRFNvqz+GN8fIw7KDeMA53Uu/GG78l945ncN579BRW8s3of76fup6y6jnEpMYzt4Tp8J7Owku5xoVTWOIgJDfS6aUYtCsLGmPOA/wC+wCuWZf3fEc9fBDwKOIE64LeWZS1t5bW2Lg9Vwo9XXv1PV20VhAP8fJjUO45PNmQ1ea5bTAiBfr5M6deBeZsO8tdLzjipO2Lnb83mb19uJTk6hBW781n6+0lHndu5fp/r7cf/rtxL+/r5mYOTIukRF8plgxP539oMUuJCee4q15HGKfFh/GJYZ/67cq/7J+qGmZIju0WRUVhJz9NoML+IiHhekL9vo+PQXX7639GwIH9mju3KdaOTWZbmOlo8MTK40QQnTwTNzlHB3Hdeb+6e3BOnRaOJJK3RFngy/eSPnMYYX+A54HygL3ClMabvEZfNBwZaljUImAm80toLbXWn2BHLBWWuIBwZ3HaboM6rny/Zrn6jxpge0fgY6NfRVZGdOiCB0uo6Fh0xIL21fbs1h1255czflkNFjYPXlu1p9rqDxVUcKK7i+tHJOJwW/5m/k/bB/nSrnx/5m3NS6BEXyhOXD2zUY3vXOT3x83H9pxAW6MdnvxrLi9cM4a0bR/L1XeO9cpeyiIjYl6+PYXzPWJKiQ7xqjKmfr89Rx/J5q5asdjiQZlnWbsuyaoB3gYsOv8CyrDLrULNxCOD95VbL6ekVHJP88mraB/u36ca0Cb1iCQ7w5edDE7lyeGf+enF/Uh+c7B6iPqZHDFEhAc1WjU/Uil353PHWOmodTjZnHTo609/X8PrydEqqat2PrdvnGjGzYb9r9NFFgzpy+dBE6pwWg7tEuoNsYmQw3949gUGdG8/5jAsP4vaJ3QkN9GPZA5PoFhvKlH6u8+5bMu5MRERETk0tSVWdgMMbJzPqH2vEGHOJMWYb8AWuqrB3asjrXhyEa+qcTH9hOct35bkfyy+rIbqNR2IFB7iqo7+b0ovHLx1AckxIo5YEf18fpg3syDdbsymuqP2RVzp2n2zI5IsfDvDR+ky2HShlWHIkvj6GBy/sS2lVHW+u2Au45lVe+vxynp6/k/X7igjw9aFvx3DunJRCO39fxvRo2cbCOyf1YNn9k05otqaIiIicWloShJuruTep+FqW9ZFlWb2Bi3H1Czd9IWNuNsakGmNSc3NP7tvpP8mLN8tll1SRureQeZsOuh/LL68mOqTp8Z8nW/fY0B8Nh5cNTqSmzumetdsaHvlsM++ucf3s9fiXW6lxOLl2VDIbHp7MdaOTmdgrlleX7qGyxsG6+gH4ry1LZ9GOXPp2DCfQz9d9str1o5NbdE9jDBHtFIJFRETspCVBOAPofNjnicBR3wu3LGsx0N0Y06QUZ1nWS5ZlDbUsa2hsbOwxL7Z1eH9FuKi+urrlwKF5f/llNW22Ue5YnNHJNf7nw3UZR73G6bRYujOPlo7qe21ZuvvjwvrvRf9Oh462vPOsHuSX1/DO6n18n1FEgJ8PZdV1bDtYypldDrU9RIUEeFXvlIiIiHiXlgThNUCKMaarMSYAuAL49PALjDE9TP2cDmPMYCAAaHrAuTfx4iBcUOHaGLclqwSn0xUe88trjjopwZOMMVw6OJHUvYXszS9v9pqFO3KYMXtVi6rGZdV17o//d+so4sICCQv0o8thQ+WHJkcxomsULy7exeo9BQzu0p4L+ycArhmGIiIiIi3xk0HYsqw64E5gHrAVeN+yrM3GmFuNMbfWX3YZsMkYswHXhIlfWJ46qaOlnN4ZhC3LYm16AQDlNQ72FlTgcFoUVtQQHdr2rREtcfGZHTEGPlyX2eS5qloH8za5ZhE39PX+mG31VfBXrx/KsOQo/n7ZAP5wYZ8mkxt+NSmF7JJqth0sZWDn9twzpRdn945jXAt7gkVERERatCXesqwvgS+PeGzWYR//Hfh76y7tJPHyzXKfbTzA0wvS3J9vySohLMgPy4IYL2yNAEiIaMeY7jF8uD6D356T0miI92NfbHEfYrFqTwELtmXz0MebefGaIZzRqelswYZ2kL4JrufO6t38YR1jekRjjOu3c0Cn9nSNCWH29cNa+0sTERGR09ipNeytNXnZZrmqWgczX1/DvM2HNsj5+Rg2ZxWTXz9D2BtbIxpcOrgT+wsqmbc5m5F/m8/ytDwcTovluw51yAT4+nD///uBzKJKZi3a1ezrbM4sISokgPjwH69+G2N468YRdIsNYWS3qFb9WkRERMQebBiEvbMinFFYwYJtOSzZcWiaRo+4UDZnlZBfXn+8sgemRrTUlH4dCA7w5a9fbuFgSRUvL9nNL19fw+5cV9/w7OuGcuGABHJKXV/LV5sOcqC4ssnrbDlQQt+E8CZHQzZndPcYFvxuote2jIiIiIh3s2EQrudlQbhhUkRJ1aHNYn07hrPlQIm7IuytrREAIYF+XNg/gf0FrnC7aEcui+tD/QX9O3B2n3hmjEwCoEN4EJZl8cYRPcO1Difbs0vpW39ynYiIiMjJZN8g7GVHLBdXNj6Q4uVrh9KvYwS5pdVsO+jqm/Xm1giAXwxzTdlr5+/baND0DWO7ATC4S3tGd4/m4jM7cW7fDryzeh+VNa7fhzqHk1+/s56aOqf7CGcRERGRk8l+58e6N8t511CLosNOZusWE8LkvvGs3O3qr12yMw8fA+2DvTsID0mKpEdcKHFhgfj6GJbszOP5qwczJMk10swYw9s3jQRg9Z4C5m4+yEfrM/nf2v10iQrmq/oDRPomKAiLiIjIyWe/INzAy1ojDq8Itw92HRzRpz4QbswoJibU+w+HMMbwxszh+BjDlgPFrNpdQP9mJkMADEuOpF/HcP4zfwfZJdWs31fkfq5rTEhbLVlERERszIatEQ0VYe9qjSg6LAhH1ld+I9r50zmqHeD9bRENOrZvR4eIICb1jmfjn8+l82EHYRzOGMPMMV3JLql2P9bO35fVfzwbP18b/rEUERGRNmffxOFlFeGSylrCAv1o5+/bqAWiX/08XW+eGHE0Qf6+P/r81IEJxNRPfOjdIYxhXaOICwtqi6WJiIiIqDXCWxRV1BAZEsAfL+xDcvSh1oD+iRHM3XwQf7/T72eWQD9f7pqcwtKdefxj+gBPL0dERERsxn5BuGGTnJdNjSiqrKV9sD9T+nVo9HhDj+2BoqYzd08HV49I4uoRSZ5ehoiIiNjQ6VdmbCkvqwgXV9YS0c6/yeMNQbhHXGhbL0lERETktGa/irB7wq3lqg634ASztlBcUUun9u2aPB4ZEsD/u200KfEKwiIiIiKtyYZB+DCWE8yPb+hqK0erCAPuObwiIiIi0nrs2xoBXtMeYVmWu0dYRERERNqG/YLw4SfKecmGuU82ZOFwWketCIuIiIhI67NfED6cl1SEf/veBgDatzs1Ds0QEREROR3YMAgfVhH2giBcVXuoKh2uirCIiIhIm7FhED6MFxyzXFpV5/44PMjeexdFRERE2pLNg7DnK8IlVbXuj9sFeMcECxERERE7sF8QbrRZzguCcKUrCM8Y2YUzu2hMmoiIiEhbsV8QPpxXVIRdrRGXnNnJwysRERERsRcFYQ9rqAiHB2mjnIiIiEhbsmEQPnxqhOc3yzX0CGtihIiIiEjbsmEQPoxXVIRdrRGqCIuIiIi0LfsFYcu75giXVNXi72sI8rffb4WIiIiIJ9k7fXnBEcsllbWEB/ljjPH0UkRERERsxd5B2CsqwnXqDxYRERHxABsG4cNbI6yjX9ZGSiprFYRFREREPMCGQfgwXjI1Qkcri4iIiLQ9+wVhb9ssp4qwiIiIiEfYLwgfzhs2y1XVaXSaiIiIiAfYOwh7TUVYrREiIiIibc2GQdh7WiOqah1U1zlVERYRERHxABsGYcDUf9ke3ixXWlV/qpx6hEVERETanP2CsAUYX9fHTs9WhEuqagE0NUJERETEA+wXhAF86yuwHq4Il1TWB2FVhEVERETanD2DsE99BdZZ59FllDS0RqhHWERERKTN2TAIW+DT0Brh4SBcXxGO0NQIERERkTZnwyAM+NRXYD0dhN09wqoIi4iIiLQ1+wVhyzqsNcKzPcLF6hEWERER8Rj7BWEAX8/3CBeU1/D0/J34+xoC/ez52yAiIiLiSfZsTvWCzXJ//2obVbWu8W3GGI+tQ0RERMSubFiKtLwiCCv7ioiIiHiWDYMwXtEjHBkS4LF7i4iIiIgdg7DlHePTyqtd9+4WE+KxNYiIiIjYmf2CMHjF+LTSqjpiwwL56rfjPLYGERERETuzaRD2fGtEaVUdMaGBBPr5emwNIiIiInZmwyDs+c1yZdV1pOWUEhZkz6EdIiIiIt7AhkGYw+YIe6YifO3sVaTnVxCuICwiIiLiMfYLwpbnK8Lr9hUBUFHj2ZPtREREROzMfkEYPL5ZLizQFcT3FVR45P4iIiIiYtsg7NmKcGJUMAAJEUEeub+IiIiI2DIIHz5H2DOtCZZlERroxwszhnjk/iIiIiJiyyBMfRA2Hp0acW7feGJCAz1yfxERERGxaxAGV3uEh4JweXUdoZoYISIiIuJR9gvCluX6fw8FYcuyKKuuIyRQQVhERETEk+wXhAEw9UG47XuEq+uc1DpcPcIiIiIi4jk2DMINFWFfj1SEy6td99SpciIiIiKeZcMgDBjjsdaIsvogHBKgICwiIiLiSfYMwuDxIKzNciIiIiKeZb8g3GizXNv3CJdV1Qdh9QiLiIiIeJT9gjDg2iznmR5hd0VYQVhERETEo2wYhD27WU6tESIiIiLewYZBGK/YLKeKsIiIiIhn2TMIg+eCsHqERURERLyC/YKwdVhrhOVs89uXV9dhDAQH+Lb5vUVERETkEPsFYeDQyXInvyJcWF7Dkp257s9Lq+sIDfDDGHPS7y0iIiIiR2fDIHz4+LSTH4TfXbOfa19dTUF5DeBqjdBGORERERHPs2EQpk03yxVX1mJZsPVACQDlNXWEqD9YRERExOPsGYShzQ7UqKxxhe0tWa4gXFpVp41yIiIiIl6gRUHYGHOeMWa7MSbNGHN/M89fbYzZWP+/5caYga2/1FZite0c4fIaV9jenFXs+ry6jjC1RoiIiIh43E8GYWOML/AccD7QF7jSGNP3iMv2ABMsyxoAPAq81NoLbV1t1xpRWR+Et9S3RpRV1xESoCAsIiIi4mktqQgPB9Isy9ptWVYN8C5w0eEXWJa13LKswvpPVwKJrbvM1tS2m+XK61sjduWWo1ZzPQAAIABJREFUU1Xr0GY5ERERES/RkiDcCdh/2OcZ9Y8dzQ3AVyeyqJOuYbOc48SD8OasYma+voaq2ub7jSvqK8IOp8X2g6WUVatHWERERMQbtCQINzfw1mr2QmPOwhWEf3+U5282xqQaY1Jzc3Obu6TttFJFeO3eQhZsy2FHdmmzz1fU1NEjLhSAzVklCsIiIiIiXqIlQTgD6HzY54lA1pEXGWMGAK8AF1mWld/cC1mW9ZJlWUMtyxoaGxt7POs9cQ0R3jcAHDUn/HLVta7T6fbklTf7fEWNg17xYYQF+rF2byFOC7VGiIiIiHiBlgThNUCKMaarMSYAuAL49PALjDFdgA+BayzL2tH6y2xtpj4I157wKzW0RKTnVTT7fEW1g5BAX/okhLNqj+vnA80RFhEREfG8n0xklmXVGWPuBOYBvsCrlmVtNsbcWv/8LOBhIBp4vv7o4DrLsoaevGWfiPqSsK9/61SE6xoqwmXNPl9RU0dwgB/9OoWzOr0AgDAFYRERERGPa1EisyzrS+DLIx6bddjHNwI3tu7STiJjTrg14tWle4gPD6K6zlUR3pN/lIpwjYPgAF96dQhzP6YeYRH5/+3deZhcVZ3/8feppfclS2ftJCQhIWwGlEVAlEVlVUEZERh3x21kXJhFUMeZ+Y0z4zj8XAdRBBUEQQXUiAoqoqDssoYlJBCSdPatk96XqjN/fG+tXd1d3ane6n5ez3OfW3XvrarTfbN8+vT3nCMiIhMvvIksGj+g0ogfPLiBuQ1V6YFw63e2470n6BEHoLc/SX/SU1sZY+WCaenjKo0QERERmXjhW2LZl6Y0orc/ycY9neke4f3d/eztzA3WqcU0quNRFs+soSEYJKeV5UREREQmXviCMJAeLJfsywTjEerpT7JlXxdt3Zkp2PLrhFOLadRWRnHOpXuF1SMsIiIiMvFCGISzeoRh1OURPf0JvIcXd7ZTFbdv4/q8mSNSi2lUB0sqH7WwEVCNsIiIiMhkEM5E5rAeYbBeYSpG/Ba9/Zn5g5fPrmfN9jZezptLuDPVI1wRBeBdJyxmRm0ls+orR910ERERESmNEPYIB1JBeIR1wl29Cbz39CYsCPclPHVVMRZOrx6wqEamR9iC8NzGKj5w8pIDbLiIiIiIlEL4grAffWnE/S/u4rDP38n9L+7OKS2ujEVY0lRbIAhbj3BNRTg73kVEREQms/AFYSA9WA5G1CP85KZ9APzy6a05x6viUZY01bF+VwfJZCYhp3qEU6URIiIiIjJ5hDQIM6ognKrt3bQnd1BcZSzCirl1dPUlaNnbBcD963bxwrY2IFMaISIiIiKTRwh/Zz/60oiKmP3csDkIu7UVUTp6E1TGoiyfYyvHrdnexqKZNVxy7UPp19WqNEJERERk0glnj7AbXWlEXzBTREurBeFlQfitikdYHqww98L2tgGvU4+wiIiIyOQTviCcHiw38iCcmikiNXVaKvxWxqLUV8VpnlbNmm1tOXXC0YijMha+b7OIiIjIZBfShOaGLI1IJv2AOmCAviAIp6SCcGpBjRVz63lhexvdwbLLADVxW1VORERERCaXkAZhIJIKwgN7hK/900u89kv3DChzSPUEpyydVUddZSw9iO6QOfW8tLMjZ9nlmkqVRYiIiIhMRiEcxTV8acQTm1oBWLOtjUOCOmCwxTOy1VXG+N1lpzC91kL1irl19CaSPLt1f/oazSEsIiIiMjmFM6W5oUsjGqvtXGtX7rn8HuGKWIS5jVXp56nQ/MTG1vQxLacsIiIiMjmFLwgXMViusdrO7c8Lwvk1wvmD4A6eVUfEwZMtFoS/fOFRvO6QWaVotYiIiIiUWEhrhLOnTxu8R3hfgSCcHX7zg3BVPMriptp0acXcxiqa6tQjLCIiIjIZhTQIM2RpRDxqszzs68w919OfTC+qAeQ8TjlsbgOtwetUHywiIiIyeYUwCA9fGpEI5gFu7co915dIUhEdOggfPr8h/bg6rhkjRERERCarEAZhhl1ZLhHUEbd2DiyNqIhF0j3G2aE45cjmxvRjBWERERGRySt8QTg9WC4ojejvGXBJIpgmLb9GuLc/STwa4d/POxKAuqqBpQ9HZPcIa2llERERkUkrpEWsDuLV9jAxMAj3JwsH4b6EJx51XHT8Ii46flHBd84eHKcgLCIiIjJ5ha9HOCVVGlGoRzgIwrs7evE+s4hGbyJJRaz4cKvSCBEREZHJK4RBOAi2zkGsCvq6BlyR6hHu7U/S0ZtIH7fBcm7YT/jga5cAEI0Mf62IiIiITIwQBmEsBAPEKgfpEc4snLGn3QbT9fYn2dLaRbzAALl8nznnMF7+4rmlaauIiIiIjInwBeGsUgdi1dDfPeCS7AXkdnVYUP78z1fzwvb2oj7COfUEi4iIiEx24QvCAGT3CBcKwgN7hP/4wk7A6oZFREREZOoLaRAOxKoKBuFUjTDArnbrEZ7dUAXA9v0DrxcRERGRqSeEQTirNCJeNeisETNqbVaJ7fvt/Jx6mxatM2vwnIiIiIhMXSEMwmQNlhu8R7g6HmVmbQXb2+x8U33lgOtEREREZOoKXxDOGSxXCX2FaoQ9kYiVQ2zf1x28zF532opZ49JMERERERlb4QvCQGawXOEe4UTSE4tEmNtQme4R7kt4muoqufY9x41nQ0VERERkjIQ0CAdig9cIRyOOOQ1V6RrhvkSSusqoFskQERERKRMhDMLZpRFV0F9oZbkksYhjdkMVu9p76E8k6UskiRWxmIaIiIiITA3hTHbDriyX6hGuxHvY2d5DX8IXtaqciIiIiEwN4Ut2WR3CQ80aEYs45tSn5g7uoS+RJB5VWYSIiIhIuQhfEAbSg+XiVUPMGuGY25hZRMOCcEi/XSIiIiJlKNzJLtUjnD2lGqlZIxyzG2zu4B37u4PSCPUIi4iIiJSLEAbhvHmE8ZDoy7miP6gRnllbSTTi2KYeYREREZGyE85klx4sV237vs6c06l5hKMRx5z6Sra2KgiLiIiIlJtwJ7uKGtvnBeFUjzDAguk1tLR20a/SCBEREZGyEr4gnF0PHK+1fW9+j7DNIwywYHo1m/d20at5hEVERETKSkiTXdCzWxEE4b6OnLP9CZs1AqB5ejVb93XR3ZugQkFYREREpGyEMNll9QinSiPyeoST3uf0CCc9bNnXrdIIERERkTISwiBMZrBcfJAe4bwa4RSVRoiIiIiUj3Anu0I9wskE0URPuke4eVp15nIFYREREZGyEb5klzNYrsCsET95L7/tvJBoxL4186ZVZWZbi6g0QkRERKRchC8IZ0sNluvNKo14bpWdcgkAKmNR5tTbUsvxWLi/XSIiIiLlJITJbpge4UBDsjX9uHm6lUdoQQ0RERGR8hHOZOfypk/L7hGungHA9MTu9KH5QZ1wXKURIiIiImUjnEE4JRKFaGVuEK5tAqCxPxOEUwPmEtn1xSIiIiIypYUvCOeH2Yqa3NKI2lkANGb1CKdKI7bt6x7z5omIiIjI+AhfEAbSK8uBzSWcPX1ajZVG1Gf1CC8IeoQ3t3aNS+tEREREZOyFMAjn9wjXQm971gELyfV9mSB8zOLpNFbH+fDrDh6H9omIiIjIeIhNdAMmhMvqEa6szw3CPgnkBuGGqjhP/ssZ49U6ERERERkHIewRzlPVAN370099sh+Aut5dE9UiERERERkH4QvC+YPlKuuhpy1zOmkLadT2KQiLiIiIlLPwBWEgZ7DcgCBsPcI1vXsgCMUiIiIiUn5CGITze4QbcoJwKvxGSEDnbkRERESkPIUwCFNgsFwbJG2QnM/uBd7XMs4NExEREZHxEs4gnK2ywfapmSOSCfb5GnvcunFi2iQiIiIiYy58QbjQYDlIl0f4ZIKX/Vw7piAsIiIiUrbCF4SBAYPlAHqCKdSSCVp9Hb2xeti3afybJiIiIiLjIoRBuMBgOcgMmEsmSBCho3q+eoRFREREylgIgzC5g+WqGm3f1QqA90EQrlEQFhERESln4QzC2Wpm2L5rj+2TCZJE6KxptiCcX1MsIiIiImUhfEE4P9imgnBnEISDHuGummabSaJr7/i2T0RERETGRfiCMJA7WK4RXCSzeEaynwQRumub7bnKI0RERETKUgiDcF6PcCQC1TNySiMsCC+w560bxrd5IiIiIjIuigrCzrmznHNrnHPrnHOXFzh/qHPuAedcj3PuH0rfzBLLHiwHVh6R6hH2SRJE6K1fZM93vzi+bRMRERGRcREb7gLnXBS4Cngj0AI84pxb5b1/NuuyPcDHgfPHpJVjrWZmpkY4mSDpIyQr66F+HuxaO7FtExEREZExUUyP8PHAOu/9S977XuAW4LzsC7z3O7z3jwB9Y9DG0io0C0RWEHbBYLmoczBzGexWEBYREREpR8UE4WYge4m1luDYFJZXGlE9PVMj7BMkiBKNOGhabj3CmkJNREREpOwUE4RdgWOjSobOuQ855x51zj26c+fO0bzF2KiZaTXC3uOCwXKRiIOZy6G7FTp2jX+bdjwPv/s3SCbG/7NFREREQqCYINwCLMx6vgDYMpoP895f470/1nt/7KxZs0bzFiVQqDRiBiR6bd7g7NKIpkPs/ESURzy3Cv70ZfjL98f/s0VERERCoJgg/Aiw3Dm3xDlXAVwErBrbZo2xAbNGzLR95x7wSZJEiESApmV2fCIGzHXvs/3v/z0zkE9ERERESmbYIOy97wcuBe4CngN+7L1/xjn3EefcRwCcc3Odcy3AZcDnnHMtzrmGsWx4SVWnVpfbnS6NiEUi0LgQYlWw64Xxb1P3PohV2/6e/xz/zxcREREpc8NOnwbgvf8V8Ku8Y9/KerwNK5mYIgbpEe7ak5k1IgJEojBrBWx/Zuyb9MBV0LET3vCv9rxnP0xbBEtPgUeuhWPeC3OPHPt2iIiIiIRECFeWKyCnNCIYLJcqn5j7Ctj29NjPHPHcHfCnr8Cmh+15936oaoBTr7BZLX7xCQ2cExERESmhcAXhwcJsTao0Yg+RdI9wKgivhM5d0LZtbNvW02b7uz5j7ezeB1WN1razvwSbH4UHrx7bNoiIiIiESLiCcEr+YLmqaRCJQbuF3aTP6xEG2L56bNvU2waVjdDyCDzzUyuNqAzKrI+8AFacA7//gpZ8FhERESmRcAXhwXqEIxFbTrnV1g3J6RGec4Tttz01tm3raYMj3wqzj4Df/avVC1cFQdg5OPfLEK2AVR9XiYSIiIhICYQrCKcVWCOkoRlaNwKQJEIsFYSrGmHaQVYnPJZ62q1n+swvQOuGTGlEun3z4Kz/gg1/gj9/bWzbIiIiIhICIQ3CBTTMTwfh/tTKcinzVsKWJ8bus/t7IdEDlXVw8OlwyNl2vK8r97qjL4Ej3gr3/Ae0/GXs2iMiIiISAiELwkPM/NAwP10jnF5ZLmXBcbB3/dgttdzbbvtUTfA5X4KKOlj2htzrnIM3fdXKOG57v80sISIiIiKjErIgHMgfLAdWGhFIZtcIAyw43vYtj4xNe3qCQFtRZ/tpi+Azm+GQMwdeWz0NLrjW6pl/9lFIJsemTSIiIiJlLlxBeKi5gBvmpx8m8ksj5h9ts0qk5vgttZ5Uj3B9cdcvOgHO+AI8fwfcd2Xhax66Bq5/C3S1lqaNIiIiImUmXEE4rYge4exe43i1TaM2Zj3CwRzClXXFv+aEj8LKi6xeeM2vB55/+V5Y/0f44YWZoC0iIiIiaSENwgVk9Qj3EyWS/51ZcDxsfgwSfaX/7Pwa4WI4B2/+Ksw7Gm77IOx4Pvd8x26oaYKWR+GWi6Gvu3TtFRERESkDIQvCQ5RG1M9NP0wQIZafhA86Cfo6YMvjpW9Wfo1wseLVcNFNtr/xAti/JXOucxcsfg2cfzWsvw9+/C6FYREREZEsIQvCgQKVEUSi6YdJn1caAbD4tbZ/6Y+j/9zHb0wv2pFjpDXC2RoXwDtvtXmHb7wAuvba8Y5d1iN81Dus53jtb1QmISIiIpIlXEF4qMFyYMGR1GC5vHO1M61OeP0og3BPO/z8Y7Dq0oHtGE2NcLZ5R1nP8K61cPMl0Nthgbhmpp0/5r3WM/zyfXDj2zSATkRERISwBeG0Ql3CwPSDgALTp6UsOcVmjshf6KIYvR22f+kPsO7uvHNBL+1ISyOyLT0F3vZt2Hg/3PR2wENtU+b80ZfAX33P6pyvfzO0bRv9Z4mIiIiUgZAG4UFMsyAco59IobmGl55qK8Bt+PPI37uvI/P4t/8MyUTmeU8bxGtzyjNG5cgLbMGNVPtSPcIpR5wPF98Mu1+E77wetq0+sM8TERERmcJCFoSHKY2YtgiA+W534R7hxSdDrBpeuGvkH93bafsj3gY7noW/fC9zrqdtdPXBhRz7PjjnSnARmHnwwPPL3wjv/zX4BHz3LFj7u9J8roiIiMgUE7IgHCjU2wuwxAbEbWf6wMFyYLMzHHyazds7XL1xvr4gCB99CSx5Hfzu/2XKE3raRl8fXMjxH4QrWmD+Kwufn3cU/M3dMGOxDaB76JqRfz0iIiIiU1y4gvBwYW/ZG7j+qJu4NXlK7spy2Q45C/Ztgu3PjOyzUzXCFbVw7legvxvuvCI41166HuGUitqhzzc2w/vuhOVnwK//EW7/YKaNIiIiIiEQriCcNkjIBbZXLyPqhvi2HHKmvf75O0b2kake4XgNNC2D1/0DPHM7rP2t9QgfyEC50aqsg4t+CKf/M6y+zeqGd60d/Prvv8m2XevGr40iIiIiYySkQXhwCe8H7w0GW3hj8cnw9E+G7mHu74Xn7sgMikvVCKd6al/zCWg6BH7xCVsIYySrypVSJGKh/J23Q8cOuOZUeOLmgV9bMgEb7rcp2K4+Ce77cmlW2Usmx2a1PhEREZFhhCwID18Hm0z6wvXB2VZeCLvXwZbHBr/mpT/Aj/4aHrvenqdmjYjX2D5WCW/9FrRvh9YNpa0RHo2DT4MP3wdzV8LPPgI/eQ907smc79hpA+xO+TSsOAvu/jf4zmlDr7S36uNw5Qp49HuQ6C98zX3/H764CO76LLRtL+3XJCIiIjKEkAXhwBBBN5Gk8IwR2Q57C0Qr4KmfDH5Natnke/7TSh/SPcI1mWuaj4FTL7fHpa4RHo3GZnjvHfCGf4PnfwXfPDEzq8T+zbafdzRceAO840Zo3wHXnAa/+CR07B74fi2P2FLPd3wSvnkCPPeLgT3NLY/YsQe/CV9baXXTmuNYRERExkE4g/AQEsnk8EG4epoNmlt96+A9nalFNzp2wp+/ltUjnDeI7eTL4Nj3w4pzDqzhpRKJwsmfhA/+Hqqnw00XwE8/khkc2DDf9oe9GT72MJzwUXjsBvjGK+HBb2XKHLyH1o1w3N/AO26yHz5+9E647gx4+U+Zz9v7Mix7PVz6qE0t99C34asr4Zd/D3teGtcvXURERMIlXEG4iCnCEt4PH4QBVr7DQu66QebhTQXhpafC/d+AHc9BJAaxitzrIlF401csDE4m81bCh/4Ar/tHq4de9Xd2PBWEwX4gOOu/4KP3w/xXwZ2fhm+dbPMsd+212TCmLYLD3gQffQDe/HWbceP758L3zrXykb0vw/TFNufxW6+Gv3sUVr4d/nI9fP1VcMtfw8YHNb2biIiIlFy4gnDa0KURBVeVy3fImVA/Dx7+duHz/UEQPvt/LACvvm1gb/BkF6+C0z8HH77Xyjjq50NN08DrZh8K7/qpzUCR6LW5ib93tp0LFikhGoNj3gMffxzO+m+rsb7hPFupb8aSzHvNWArnXQWfWg2vvcx6j797Jlz7Blh9+/AD67r32TzPmgpOREREhhGyIFzkYLlivivROBz3AXjx97BzzcDzfd22n7nMpieD3PrgqWTOEbYAxyeetFkmCnEODj3XyiXO/XJmoN3MZbnXxavhhI/Ye539P7DoRFh62sD3q58Lr/88XPasrZTXtQdufR985Uj4/RegdVPhdjz0bbj5IrjyEPjZ31qQTiZH/7WLiIhI2QpZEA4MNVjOFzFrRMox74NopYWvfH2dNqAuErGV3pqPgdoCvalThXMDyzoKSf2A8Ikn4AO/hdmHFb4uXgWv/hC8/87CS0GnVNTa9+/SR+HiW2xVvHuvtIF1N11ovb+pKerAapnr58GRb4NnV1kZxtePhnv+C/asH9nXLCIiImUtNtENmGySyWHmEc5W2wSveDs8ebOVENTMyJzr77beT7A64Hf9FHraS9/gyaqiFhYeX7r3i0Rhxdm2tW60AXqP3WC9vw0L4OiL4aiLrXd+3tHwlm9YCcbzd8ATN8Ef/xv++EVoPtZC8uHn2ywZIiIiElrh6hEuYsBVf7LIwXIpJ11qA+Pu/3ru8b4uiFVnnlc1KniVyrRF9oPHp56BC38Asw6xXuJvvAp2PgezVth1FTU25/O7fw6ffBpe/y9Ww3zXZ+Arh8N1Z1pvfjHTte3fAk/80EK4iIiIlIWQ9ggPUxoxkiA8+zB4xV9ZoDrhY1A3y473d9uv/2XsRONw+Fts27cZnvoRrP2N1Srnm7bQBt+99jJbIvrZn8Lqn8Kv/wl+/WmrVT70HJvGrlCpxn1fhke+Y49nHQrLz7Bt0QnWDhEREZlywtUjXKqV5fKdcrkF3z9/NXOsrzOzipyMvcZmC7nvv3P4koymZTYt3N/eb4P7Tvm0LYDym89Zr/L/Hge//bxN25aqP976hK26d8Z/QN0cePBquP5N8KWl8ON3w+M3WRgXERGRKSOcPcJDriw3wh5hsGB11CXw8HdsoNiMpTZrREw9wpPerBVw2hW27d0AL9wJa34FD1xlC6HUzITlZ8K21XDMe60U5qRLbbXAl/5gPdBrfwvP/tzeb+ZyWHoKLDkFlrzWFiURERGRSSmcQXgISe+Lm0c43+mfg2d/Zr9mv+THuYPlZGqYfhC8+sO2de+zxVLW/BrW/NLmhc7uaa6st9X1Dnuz1Z5vXw0v/dHC8RM3wyPXAg7mH22heOkpsOB4qKwrTVu9hxveYgMwm4+BBcfafsbBg09xJyIiIjnCFYSLWVluND3CAA3z4NQr4DeftfDU1wlV00bRSJkUqhrhyAtsS/TDnhett7cQ52DuK2w76VLo74XNf4H1QTB+4H+tbMZFbcW+RSdZbfGiEzM15YXs3QC3f8gWHFlwLCw4DmYfYYuT7FoL6++1eZqf+GGmfrmq0Vb5az4ms9XPGfwzbn2/DQCcuzL4GlbCnMP1Q5yIiIRCuIJw2uBBt38k06fle/WH4fEb4Zd/b2Glft4o2yeTSjSWmYmiGLEKOOhE20693HptNz4IGx+w7ZFr4cGr7NqZyy0UH3QSLHy1ldWkfiPx/C9h04O2Ct+TN9uxeA3Mf2UmqF58i71m5xoL35sftf2fvgI+qG9uWGA90/OOymz1c20mjNW32RLXT98Kj15n17sINB2SCfdzV9pWO/OAv5UiIiKTSciCcBGD5bwnNtogHI3D+VfBdWdAst+CjUhlHSx/g20A/T2w5QnYeL8F5OdWweM/sHPVM6znd8Fx8NI90LgIPvkUtG6AlkeD7RHY9DBMO8h6hJ2zXtw5h8Or3mXv09sJ256y6zf/xR4/f0emTbWzMz3FF95gQbd1A2x72ratT8GGB+Dpn2ReUz/fPmP2YTDrMFtae9ahNme0iIjIFBSyIBwYbrDcaGqEU5qPsVkI7vkPDZaTwmKVsOjVtoEtAb3zOQu3qaC79i47t/Ii+/M6fbFtr/grO97XDT45+J/lipqg/OKEzLHu/VbLvPUp2PqkheMFx8GcV+R+xmFvzrymc08Qjp+y/Y5nYf19kOjJXDPtoCAcH5rZz1qh8goREZn0whmEh5BMlmCs0cmXWQ3nwaeXpE1S5iIRmHOEbce+z451tVr4nH144deMZo7qqgYrwTjopOJfUzPDBvotPSVzLNEPe1+28L4j2HY+D+vuhmSfXeMiFqpnLrde66Zl9rhpuU0/dyA/bIqIiJRIuIJwMYPlvCd+oEk4GoMLvnNg7yHhVj0NlrxuoltRWDRmwbZpWW7vcaIPdr+YCcg7n7fFS9b/0WZRSamot0VLmpYH4TgIyTMPVpmFiIiMq3AF4bShSyNGNX2aSNhF41Y3PPtQOOKtmePJJOxvsd+S7F5n2661Vh+dXYMMVoc8YwlMX5Ip1ZgRPK6ZOXRPcqLPFkXB2TLc2Vu1ZnAREZGBQhaEixssN6rp00SksEgkE0iXvT73XG8n7HkJdqdC8otWdvHi3dC2NffaivogGC8OQvKSTFBuXAgv/h4e+pbV5mf3QANUNg4MxwrKIiKhF7IgHBiiV6k/cQCzRojIyFTUwNwjbcvX22lzHO9dD3vWW0Deu96minvhN7kD9lzEpparbIR/XAu9HTYLRuvG3G3veivV6G3P/axUUG5cAA3zbUs/bra9Bv+JiJSdcAbhIYx6ZTkRKa2KmkypRb5k0nqM96YC8ssWdA86yWbliFXaQL/5rxz4Wu+ha2/hoLyvxeZu7to78HXVMywUNzZnwnJDXliuqCn8tdz5GXjmpzZlXd1c29fPs4GD9XNtq5sLtbOsBltERMZFuP7FHcuV5URk/EQiFkgbm2HxySN7rXMWkgcLymC90W1bLRjv3wL7Nwdb8HjTw9C1Z+DrqqdbwK2fm9nXzbGV/2YdanXO+1ps4ZOOnQXaFrEwnB+Q08E5eFw7y8K+iIgckHAF4bQhBsv5A1hZTkTKQ0WNzWIx8+DBr+nrygrJWzKhuX27hegdz9tjnwAcnP9NW6kvJdEH7TugfRu0BVvqtW3BfuuTFph9cuDnVzZCbRPUzbZ97SxbKCX1uG52cGyWLb2t33SJiAwQ0iA8uOSBLqghIuEQrx4+LCcT0LELEr0wbWHuuWg806s9lES/heH2bRaQ27fZ845dFqQ7dtosHBvutwVQCg0KjsSDcDwrE47TW5P1VFcHveQ1MxWcRSQ0whmEh1pZTrNGiEipRKKZpayirYBCAAAMZUlEQVRHKxqDhnm2DSfRD527g6C8Izcsd+wKju203uqOnbkDDnPaHbMyj5qZwTYjCMozc49l7ysbCv/buvkxWPVx+8Ghevog27Tc51WN9r0TkdJJJuCBq+zfgXit/earojbzOL2vgYq6rMe19oM7wNrfwZM326JOBV9XG+xrYPVt8OSPMs9T+0t+Uty/Z+MknEF4CImEBsuJyBQVjQX1xEWEb++hZ78F5869wT7YuvZkPd9rC6OknvtE4feLxLLC8oxMqN3yhA1EnH90UDLynA1G7G0bun1VjXnheNrQQbqq0a6JVw/dm71rLdx7pf1HXtVoAb6qMbOlnwf7irpw9I4ng/KbA15atYT2rLfvfUWdbbFKW+Z91cdtFcuKOgte6a1umMd5+3jN5Pp6S6W30wYQZ4fUWDX84Ytw75cKTzE5nEjcvme97fb3IloJfR32WakVRQs54m32m6fUtX2dk258Q7iCcJEry2n6NBEpe85lwt+MIl+TE5735O2zQnTHbpsfumsvdO+D0/8ZTvzb3PdK9Nm5rr0FttaBx/ZugO7geKGa6ZRIPPN1pQNyYyYov3wfbHvannfvs7KVIb9PkSAcB8G4sjE3KOcH5/S10zKPS/0fv/cjC+cdu+D5OzI9fZV1UFlvc3NX1oGLwnVvDMJTbeaaitrMNanwWFlf/Pl47eiC5ubHbJaV+7+ee9xFg/tRDwuPt2kSO3ZaYO7tCLb2wX9YKyQ+WJDOel6ZH6TzwnT+PlY5/P1Zd7etvpn9unh1gfcMelJTPbJg35tnfx68Juua1OMHvgl7Xiz8uUe/E86/yn7w6eu071l2SE19H9PnOu1cb7s9jlbAyZ+ykqqURF/utan3i1XagORJ/oNkuIJw2lAry6HBciIiheSE56UH9l7ReDCwr2n4a7Mlk9abnB+Uu/dbsO1uDfbB1tUKrZsy5xK9cOZ/wokfs/fr67Zw370veI/W4HlwLOdc8Lx1Q+7z4RZrilZYeEuHz+wtCKWVDVkhMv+a+sy5TQ/DD863QJi+ti7rfety36OiDtb8yn4AGM6Jl1rI7m2DnvZMsNy/xfapY30dxd+veO3IgnTrRvjzV+21y8+Aw88P2tEW7DvhqHcMPuOL99Dfk2l7dkDOeTzEue5WGwSbfW64H5iyuUhWuUD1wNKBaNyC7EhE4pke3vbt9puXaEUmdGaXOcWq4Jwr7fOyA228Go7/cPB+keDPXt3I2lFINB6UN03NhYlCFoSLXVluHJoiIiIjF4lkwvj0xSN/fX8vxCoyz+NVttXNHl17UsG8e3/h0NzdGgTIduhpy2zt2201xdTz/q7iPs9FbBq9I99mr0u/b7stZZ563Nue++vv0z5rv6ZOhdz0a/fb82kL4cgLivyaE7khMf1e7ZmQ2dM++LG2rbC7I3MsP1gvfq21d8FxI59X27nMPa2dObLXDqW/Ny8wd2R6SQv1hg52vGOXPV72RptJJtk/8HWF3iv7WEUdnHpFbvBMJjLXpMp+pCghC8KBoQbLadYIEZHylR2CSyE7mB+IRH8QUtsyYbanLQiqbbmh97A3w6ITinjPviBkd9s81KX6vy0SDco/GkrzfslkbqicvnjyDZaMVUAsmFllMopEM78FkBEJZxAeQjKpeYRFRGScRWOZwX8le8/45A1u2SKR0gZrkREIVxFAEYPl+tUjLCIiIhIK4QrCRUh4TzSqICwiIiJS7hSE82hlOREREZFwCFkQLm4eYa0sJyIiIlL+QhaEA4P0+Hrv8R6tLCciIiISAuEKwsMMlksk7bx6hEVERETKX7iCcFrhoNuvICwiIiISGiENwoUlgx5jlUaIiIiIlL+QBeHiSiNi6hEWERERKXshC8KBQXp8k0nba2U5ERERkfIXriA83GC54LzW0xAREREpf+EKwmmFk65mjRAREREJj5AG4cLSg+UUhEVERETKXsiC8NClEenp0zRrhIiIiEjZC1kQDgw6WE6lESIiIiJhEa4grJXlRERERCRQVBB2zp3lnFvjnFvnnLu8wHnnnPt6cP4p59yrSt/UUhpksJxXEBYREREJi2GDsHMuClwFnA0cDlzsnDs877KzgeXB9iHg6hK3c1ykSiO0spyIiIhI+SumR/h4YJ33/iXvfS9wC3Be3jXnATd48yAwzTk3r8RtLYEi5xFWj7CIiIhI2YsVcU0zsCnreQvw6iKuaQa2HlDrSmxvRy/TgWv//DL3PvnwgPPt3X2AeoRFREREwqCYIFwoFeZ3rRZzDc65D2GlEyxatKiIjy6tRCTOmtgKtvfXsb+rr+A1Jy6dyZHNDePcMhEREREZb8UE4RZgYdbzBcCWUVyD9/4a4BqAY489dug6hTHQNHchTZ97mM+O9weLiIiIyKRTTI3wI8By59wS51wFcBGwKu+aVcC7g9kjTgD2ee8nVVmEiIiIiEi2YXuEvff9zrlLgbuAKPBd7/0zzrmPBOe/BfwKOAdYB3QC7xu7JouIiIiIHLhiSiPw3v8KC7vZx76V9dgDHytt00RERERExk64VpYTEREREQkoCIuIiIhIKCkIi4iIiEgoKQiLiIiISCgpCIuIiIhIKCkIi4iIiEgoKQiLiIiISCgpCIuIiIhIKCkIi4iIiEgoKQiLiIiISCgpCIuIiIhIKCkIi4iIiEgoKQiLiIiISCgpCIuIiIhIKCkIi4iIiEgoOe/9xHywczuBDRPw0U3Argn4XBlfus/hoPscDrrP5U/3OBwm8j4f5L2flX9wwoLwRHHOPeq9P3ai2yFjS/c5HHSfw0H3ufzpHofDZLzPKo0QERERkVBSEBYRERGRUApjEL5mohsg40L3ORx0n8NB97n86R6Hw6S7z6GrERYRERERgXD2CIuIiIiIhCcIO+fOcs6tcc6tc85dPtHtkdFzzi10zt3jnHvOOfeMc+4TwfEZzrnfOufWBvvpWa+5Irj3a5xzZ05c62WknHNR59zjzrk7gue6z2XGOTfNOXerc+754O/1ibrP5cc596ng3+zVzrmbnXNVus9Tn3Puu865Hc651VnHRnxfnXPHOOeeDs593TnnxqP9oQjCzrkocBVwNnA4cLFz7vCJbZUcgH7g7733hwEnAB8L7uflwN3e++XA3cFzgnMXAUcAZwHfDP5MyNTwCeC5rOe6z+Xna8Cd3vtDgaOw+637XEacc83Ax4FjvfdHAlHsPuo+T33fx+5RttHc16uBDwHLgy3/PcdEKIIwcDywznv/kve+F7gFOG+C2ySj5L3f6r1/LHjchv2n2Yzd0+uDy64Hzg8enwfc4r3v8d6vB9ZhfyZkknPOLQDOBa7NOqz7XEaccw3A64DrALz3vd77VnSfy1EMqHbOxYAaYAu6z1Oe9/5eYE/e4RHdV+fcPKDBe/+At8FrN2S9ZkyFJQg3A5uynrcEx2SKc84tBl4JPATM8d5vBQvLwOzgMt3/qeurwD8Byaxjus/lZSmwE/heUAJzrXOuFt3nsuK93wxcCWwEtgL7vPe/Qfe5XI30vjYHj/OPj7mwBOFCdSaaLmOKc87VAbcBn/Te7x/q0gLHdP8nOefcm4Ad3vu/FPuSAsd0nye/GPAq4Grv/SuBDoJfow5C93kKCmpEzwOWAPOBWufcO4d6SYFjus9T32D3dcLud1iCcAuwMOv5AuxXMjJFOefiWAi+yXt/e3B4e/DrFYL9juC47v/U9BrgLc65l7FyptOdczei+1xuWoAW7/1DwfNbsWCs+1xe3gCs997v9N73AbcDJ6H7XK5Gel9bgsf5x8dcWILwI8By59wS51wFVqi9aoLbJKMUjCS9DnjOe//lrFOrgPcEj98D/Dzr+EXOuUrn3BKsCP/h8WqvjI73/grv/QLv/WLs7+zvvffvRPe5rHjvtwGbnHMrgkOvB55F97ncbAROcM7VBP+Gvx4b36H7XJ5GdF+D8ok259wJwZ+Pd2e9ZkzFxuNDJpr3vt85dylwFzZS9bve+2cmuFkyeq8B3gU87Zx7Ijj2GeCLwI+dcx/A/tF9O4D3/hnn3I+x/1z7gY957xPj32wpEd3n8vN3wE1BR8VLwPuwjhrd5zLhvX/IOXcr8Bh23x7HVhmrQ/d5SnPO3QycCjQ551qAf2F0/05/FJuBohr4dbCNffu1spyIiIiIhFFYSiNERERERHIoCIuIiIhIKCkIi4iIiEgoKQiLiIiISCgpCIuIiIhIKCkIi4iIiEgoKQiLiIiISCgpCIuIiIhIKP0fp38q0yUHExYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "nn = np.arange(1, n + 1)\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(nn, np.cumsum(X1) / nn, label='p = 0.3')\n",
    "plt.plot(nn, np.cumsum(X2) / nn, label='p = 0.03')\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.21 (Computer Experiment)**. Suppose we flip a coin $n$ times and let $p$ denote the probability of heads.  Let $X$ be the number of heads.  We call $X$ a binomial random variable which is discussed in the next chapter.  Intuition suggests that $X$ will be close to $np$.  To see if this is true, we can repeat this experiment many times and average the $X$ values.  Carry out a simulation and compare the averages of the $X$'s to $np$.  Try this for $p = .3$ and $n = 10, 100, 1000$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from tqdm.notebook import tqdm\n",
    "\n",
    "B = 50000\n",
    "p = 0.3\n",
    "\n",
    "np.random.seed(0)\n",
    "\n",
    "X1 = np.empty(B)\n",
    "X2 = np.empty(B)\n",
    "X3 = np.empty(B)\n",
    "for i in tqdm(range(B)):\n",
    "    x1 = np.where(np.random.uniform(low=0,high=1,size=10)<p,1,0)\n",
    "    x2 = np.where(np.random.uniform(low=0,high=1,size=100)<p,1,0)\n",
    "    x3 = np.where(np.random.uniform(low=0,high=1,size=1000)<p,1,0)\n",
    "    X1[i] = np.sum(x1)\n",
    "    X2[i] = np.sum(x2)\n",
    "    X3[i] = np.sum(x3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X1 mean: 3.010\n",
      "X1 np:   3.000\n",
      "\n",
      "X2 mean: 30.013\n",
      "X2 np:   30.000\n",
      "\n",
      "X3 mean: 300.104\n",
      "X3 np:   300.000\n"
     ]
    }
   ],
   "source": [
    "print('X1 mean: %.3f' % X1.mean())\n",
    "print('X1 np:   %.3f' % (0.3 * 10))\n",
    "print()\n",
    "print('X2 mean: %.3f' % X2.mean())\n",
    "print('X2 np:   %.3f' % (0.3 * 100))\n",
    "print()\n",
    "print('X3 mean: %.3f' % X3.mean())\n",
    "print('X3 np:   %.3f' % (0.3 * 1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "ax = plt.subplot(3, 1, 1)\n",
    "ax.hist(X1, density=True, bins=100, label='histogram', color='C0')\n",
    "ax.vlines(X1.mean(), ymin=0, ymax=5, label=r'$\\overline{X}$', color='C1')\n",
    "ax.vlines(0.3 * 10, ymin=0, ymax=5, label=r'$np$', color='C2')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set_title('n = 10')\n",
    "\n",
    "ax = plt.subplot(3, 1, 2)\n",
    "ax.hist(X2, density=True, bins=100, label='histogram', color='C0')\n",
    "ax.vlines(X2.mean(), ymin=0, ymax=0.5, label=r'$\\overline{X}$', color='C1')\n",
    "ax.vlines(0.3 * 100, ymin=0, ymax=0.5, label=r'$np$', color='C2')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set_title('n = 100')\n",
    "\n",
    "ax = plt.subplot(3, 1, 3)\n",
    "ax.hist(X3, density=True, bins=100, label='histogram', color='C0')\n",
    "ax.vlines(X3.mean(), ymin=0, ymax=0.05, label=r'$\\overline{X}$', color='C1')\n",
    "ax.vlines(0.3 * 1000, ymin=0, ymax=0.05, label=r'$np$', color='C2')\n",
    "ax.legend(loc='upper right')\n",
    "ax.set_title('n = 1000')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2.10.22 (Computer Experiment)**.  Here we will get some experience simulating conditional probabilities.  Consider tossing a fair die.  Let $A = \\{2, 4, 6\\}$ and $B = \\{1, 2, 3, 4\\}$.  Then $\\mathbb{P}(A) = 1/2$, $\\mathbb{P}(B) = 2/3$ and $\\mathbb{P}(AB) = 1/3$.  Since $\\mathbb{P}(AB) = \\mathbb{P}(A) \\mathbb{P}(B)$, the events $A$ and $B$ are independent.  Simulate draws from the sample space and verify that $\\hat{P}(AB) = \\hat{P}(A) \\hat{P}(B)$ where $\\hat{P}$ is the proportion of times an event occurred in the simulation.  Now find two events $A$ and $B$ that are not independent.  Compute $\\hat{P}(A)$, $\\hat{P}(B)$ and $\\hat{P}(AB)$.  Compare the calculated values to their theoretical values.  Report your results and interpret."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "np.random.seed(0)\n",
    "\n",
    "B = 10000\n",
    "results = np.random.randint(low=1, high=7, size=B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "A_hat = np.isin(results, [2, 4, 6])\n",
    "B_hat = np.isin(results, [1, 2, 3, 4])\n",
    "AB_hat = np.isin(results, [2, 4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "nn = np.arange(1, B + 1)\n",
    "\n",
    "\n",
    "f, (a0, a1) = plt.subplots(2, 1, gridspec_kw={'height_ratios': [3, 1]}, figsize=(12, 8))\n",
    "\n",
    "a0.plot(nn, np.cumsum(A_hat) / nn, label=r'$\\hat{P}(A)$')\n",
    "a0.plot(nn, np.cumsum(B_hat) / nn, label=r'$\\hat{P}(B)$')\n",
    "a0.plot(nn, np.cumsum(AB_hat) / nn, label=r'$\\hat{P}(AB)$')\n",
    "a0.plot(nn, np.cumsum(A_hat) * np.cumsum(B_hat) / (nn * nn), label=r'$\\hat{P}(A) \\hat{P}(B)$')\n",
    "a0.legend(loc='upper right')\n",
    "\n",
    "a1.plot(nn, np.cumsum(A_hat) * np.cumsum(B_hat) / (nn * nn) - np.cumsum(AB_hat) / nn, \n",
    "         label=r'$\\hat{P}(A) \\hat{P}(B)- \\hat{P}(AB)$', color='purple')\n",
    "a1.legend(loc='upper right')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our own choice of non-independent events, let $A = \\{ 2, 4, 6\\}$ and $B = \\{2, 4, 5\\}$.  Then $\\mathbb{P}(A) = \\mathbb{P}(B) = 1/2$ but $\\mathbb{P}(AB) = 1/3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "A_hat = np.isin(results, [2, 4, 6])\n",
    "B_hat = np.isin(results, [2, 4, 5])\n",
    "AB_hat = np.isin(results, [2, 4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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sWrRo1GWeqTPumMMYEwQ2Au8E9gMvGGO+b639zXGT3QL8xlp7gzGmFnjVGLPZWps50/VPKKPeEUVERETkje/whg2kt79S0GVGlyym/rbbTjtdU1MTixcvxlrLfffdx/XXX08mk2H37t3MmzcPgObmZrZt28aDDz7Is88+S3NzM5dddhkAF154IS+88MK4lje4zJtvvplsNsv+/ft57LHHKCsrG3WZZ6oQNWGrgZ3W2t0DoerbwI0jprFAqTHGACVAB1C4RpUiIiIiIjLttbS00NfXxzXXXMPq1avp7Oxk48aNtLW1UVFRMTTd7bffzt13340xhiVLlgx79lcwGCQSidDb2zvm5bW0tDBjxgwaGxvZvn07Gzdu5O677x51mYVQiC7qZwEtx73eD1w6Ypp/BL4PHARKgQ9Ya/3RFmaM+QTwCYA5c+YUoHgiIiIiIjIeY6mxOhsaGxtZu3Ytjz/++LDhvu+TSqUAeP7553niiSfYunUrt9xyC6lUimXLlg2bPp1OE4vFxrS8wfUuXbp06PXy5cv5yle+MuoyC6EQNWGjtdEbeePUNcBW4BxgBfCPxpiy0RZmrf2GtXaVtXZVbW1tAYonIiIiIiLTQVNTE8uXLz9heGVlJZ7nkUqluO2223j00Ud5/fXXef3113n55ZeH1YS1t7dTW1tLOBwe0/IG1zt435i1lgcffJCrrrpq1GUWQiFC2H5g9nGvG3A1Xsf7KPBd6+wE9gCLC7BuERERERF5g2hqajqhVmvQ1VdfzTPPPEM6nWbt2rVDw+vq6kgkEnR0dADw1FNPcd11141pec8+++zQdJs2bWLlypVcfPHFpFKpYc0Rj19mIRSiOeILwHnGmPnAAeD3gA+NmGYfsBZ4xhhTB5wP7C7AuieeOkcUERERETkrTtUD4fr167n33nuHgtPxuru7h/7esmUL99xzz5iXd9VVV52258Pjl1kIZxzCrLU5Y8x64AkgCDxgrW02xvzxwPivA3cDm4wxTbjmi5+11rad6bonnnpHFBERERGZDCtXruTKK6/E87yhZ3uNlMlkWLduHeeff35BljfeZY6VsXbqVu2sWrXKvvjii5NdjGPuvQAWXAHrNk52SURERERECmr79u1D90XJ+Iy274wxL1lrV402fUEe1iwiIiIiIiJjoxA2Dp61PL+nnVTWm+yiiIiIiIjINKUQNg7dySx72/t5cvvRyS6KiIiIiEjBTeVblaaqfPaZQtg4GSxG/XOIiIiIyBtMLBajvb1dQWwcrLW0t7eP+yHOheii/rfG4MdRGUxERERE3mgaGhrYv38/ra2tk12UaSUWi9HQ0DCueRTCRERERESEcDjM/PnzJ7sYvxXUHDEPao4oIiIiIiL5UggTERERERGZQApheVFVmIiIiIiI5EchbJzUFFFERERERM6EQtg4BfAVxEREREREJG8KYeNQnTvKTcFnCac7JrsoIiIiIiIyTSmE5SGWPDrZRRARERERkWlKIUxERERERGQCKYTlwRjtNhERERERyY/ShIiIiIiIyARSCMuHukcUEREREZE8KYTlRSFMRERERETyoxCWB1WEiYiIiIhIvhTCREREREREJpBCWD5UFSYiIiIiInlSCBMREREREZlACmF5MOqYQ0RERERE8qQQlg81RxQRERERkTwphOVDGUxERERERPKkEJYHqxQmIiIiIiJ5UgjLg+4JExERERGRfCmE5UP3hImIiIiISJ4UwvKhECYiIiIiInlSCMuDsZNdAhERERERma4UwvKiFCYiIiIiIvlRCMuDtQphIiIiIiKSH4WwPFjVhImIiIiISJ4UwvJgVBMmIiIiIiJ5UgjLg7X+ZBdBRERERESmKYWwPKgeTERERERE8qUQlg9VhImIiIiISJ4UwvKgjjlERERERCRfCmF5UBf1IiIiIiKSL4WwvCiEiYiIiIhIfhTC8qCaMBERERERyZdCWD4UwkREREREJE8KYSIiIiIiIhNIISwPao4oIiIiIiL5UgjLh0KYiIiIiIjkSSEsD1ZPaxYRERERkTwphOVBFWEiIiIiIpIvhbC8KIWJiIiIiEh+FMLyoI45REREREQkXwphIiIiIiIiE0ghLA+hTO9kF0FERERERKYphbA8XPCbr0x2EUREREREZJpSCMtDMJec7CKIiIiIiMg0pRAmIiIiIiIygRTCxuFQMMh9leVYdVEvIiIiIiJ5Uggbh1tn1PC/K8rZFVQIExERERGR/CiEjUPaGAD8SS6HiIiIiIhMXwph42AmuwAiIiIiIjLtKYTlRc0RRUREREQkPwphedp2oJvORGayiyEiIiIiItOMQlierv+HZ/mdr/5ssoshIiIiIiLTTEFCmDHmWmPMq8aYncaYz51kmiuMMVuNMc3GmP8qxHon2+vt/ZNdBBERERERmWbOOIQZY4LARuBdwFLgg8aYpSOmqQC+CrzHWnsB8L4zXe+ksiPuCTu4Fb63Hnz1mygiIiIiIqdWiJqw1cBOa+1ua20G+DZw44hpPgR811q7D8Bae7QA6500mVDJ8AHfuBx+/S+QmNabJSIiIiIiE6AQIWwW0HLc6/0Dw463CKg0xjxtjHnJGHPzyRZmjPmEMeZFY8yLra2tBShe4e2pfcfoI6xqwkRERERE5NQKEcJGe3zWyD7cQ8DFwLuBa4AvGGMWjbYwa+03rLWrrLWramtrC1C8whnc0H0dJ7kXLJeasLKIiIiIiMj0FCrAMvYDs4973QAcHGWaNmttAkgYY34KLAd2FGD9E+6kHXLk0hNbEBERERERmXYKURP2AnCeMWa+MSYC/B7w/RHTfA94uzEmZIyJA5cC2wuw7klykoc1Z5MTWwwREREREZl2zrgmzFqbM8asB54AgsAD1tpmY8wfD4z/urV2uzHmcaAR8IF/stZuO9N1T5bR2l8CqgkTEREREZHTKkRzRKy1PwB+MGLY10e8/jLw5UKsb7KZk9WE5VQTJiIiIiIip1aQhzX/tgmoOaKIiIiIiORJISwPVfHwsRfHP7j52x+a+MKIiIiIiMi0ohCWh1Dw2F1hVt3Si4iIiIjIOCiE5eGm/keG/s6lT9JdvYiIiIiIyCgUwvIQNP7Q316qbxJLIiIiIiIi001Bekf8bXF81/QfC/6QdwWfJ5d+cNLKIyIiIiIi049qwsZhyWvwnXty2Kzhi+F/4ZLADvxEx/CJDvxqcgonIiIiIiLTgkLYOFzx84G6sN5ju61sy7uHT3To5QkskYiIiIiITDcKYYVSUuf+r5w3qcUQEREREZGpTSGsUC56n/vfy05uOUREREREZEpTCCuU+Ze7/3/8xckth4iIiIiITGkKYeNgh7pHNCeOLKp0/7dun6jiiIiIiIjINKQQlgc74vVW/1z2xxa4F3PeMuHlERERERGR6UMhLA8jQ9jD3hWkbBTiNTBjMez5KXTtG/dy33LPk8z73GOFKaSIiIiIiExJCmHjMNgI0Y5ojZi2YfozOYgUQyYBD94Af38R2JFx7eR+uaeDg90pAPrSuQKVWEREREREphqFsAJIE+FLj7/iQti27x4bkUuNeRn7OvqH/l71Vz/G88ce4EREREREZPpQCMvDyHiUJkxHIgvhOPjHdVGf7j3pMp569SiPNR4aev29rQeG/k5lfZZ84fFCFVdERERERKYQhbACyBDiivNr4cCLw0f84yUnneej33yBW7b8CgBrLc+81jZ8mZ5f8HKO5PuWVNY76+sREREREZFjQpNdgOloZE1YlhBlsfCJE6a6Rp3/y0+8MvT3yI44/mrdhdzxf7cBkPN8DnQlOaeiiHBwbHk5mfH4yT/8Ma9GLuB/rP/vGDP8BrZU1uPnu9v56DdfGBr2J1ecywdWzWZeTfGY1iEiIiIiIvlTCMuDHdEzh28DfOnxV/iT2Njm3/jUrlGH/+k7FvL7l83laG+a+558jYW3/3Bo3M6/fhehYIBf7+tkT1uCm97UMGzeTM7Ht5a/+MvP8zeh7/Bu4KrbithpG9i14TqCAVfmxaM0c/za07v42tO7+Iv3XMB/e8u8sW2EiIiIiIjkRc0Rx8GO+H+QP9Bv4j/mbjw2cO5bYe7bxrX8P3/nIgAuW1B1wriFt/+QjkSG3/nqc/z5d17mbV/6ybDxH//il/gfd/4FfxO6f2jYf0Y/A8C5t/0Aay0bfnDig6SLI8Ghv//1F3vHVd58/OSVI/yo+TDzP/8Y8z73GHvbE+qEZJLlJqDp63Tk+5b+TA7ftyQzarYrIiIihaOasHyM6KJ+MEKsD33v2MBYOXS1jDr7jNIoR3vTLJlZxvZDPQD8+gvvHGo6uKS+bNT53nT3j4f+3t+Z5JLPbea7q5qoyx7kW5HvjTrP67EP8Zo/i/mfPzbsn25exVVL64Zed/dn+eSWl/jZzna+9fPXufnN80Zd1plo6ejnd776HG196WHDL//y08Ne15RE+PrvX8x7v/7zoWGfvOJcPnPt4qHX2w5087X/2sUXr19KXdkYqx/f4FJZjy3P76Ols5+eZI72RJpVcyuZXRXnHYtn8MNthznSnWJXax/bDvawtz1BKBAgnfPwLQQMBAOGurIY1SVR5lfHefeyc5hfE2dBTQmBgMH3LYHAsQ+/71t+saedXa0Jls4sY3F9KQFjCAcNOd8SCweHpvvRb47wm4PdHOpOURILsbyhgkQmxyuHemk+2E0oGGBmeYxYKEgy67GoroSb3tTAORVFZ33feb7lZzvbONqb5kfNhznQlSQWDvLrfZ0cf32gIh7mbQtr+IO3zGPVvBMvlIiIiIiMlbHjeJbVRFu1apV98cUXTz/hBPnJ2xczs9XQf0MXFxcf61J+Xfov2WoX8l+RTzE3cBQAa4IY68Hn9rlANsBay/zP/4CPvnUed95wAb/Y3U44GODiuZXD1uX5lr/78Q7m1RTzRPNhfvybIyNKY3k99uHRC3rbQY788G+p+/XfDw26Kv231JpullbBFz79mRNm+UHTIT652XUU8uYF1Tz0icvGs2tOylpL4/5ubtz4s4Is71S2fPxSVs+vIjTG++emop5Ulmd2tLFiTgUvt3Sx7UA3l8yr4pL5Vfx0Ryu/OdiDZy1fe3oXb5pTQXE0dEKnLqdiDJw3o4RIKEBtSRTfwqzKItr70iTSHh2JDK8c7hkWPhbUFJPKehzuSbH0nDIW15extaWLnUf7Rl2+tRAJBVg1t5LW3jT7OvpJ51xtWzhoyHrHFh4NBWiodPc8HuxK0pMa/oy8mpIIc6uL2d3aRzgYIBIKEDCGOVVx1iyqYU9bPz/d0UosHGD1/CrCQTe+Mh5hycxS1i6pIxgw9KVz7O/spy+VY2tLFy/t7SRgDMmsx893tZMc6KAmEgzQUFVEOBDgTXMrKAqHyHo+pbEQTQe6+fW+LvrSOc6tLaa6OEosEmRGaZRlDeXMrykmmfGIR0LMqiyivCjM0d4UnYksO4/2UlUc5Z1L64iEhn8+u/oz/OZgDyvmVOD5lkzOp7oketL30PctB7qStPalae/L0NqbpiuZIZXxmFdTzGULqjmnooic55/wXUhmPA50JVlQUzwUqBPp3MC+zxIMBEikc1w6zb9HIiIiU4Ex5iVr7apRxymEjd3JQti81GbAYPDZHv0oH8zcwb9H73Qj3/EFWPNpwAWS77zYwmf/rYnZVUU885l3jHnd1/zdT3n1iOvy/rIFVXz74LUnTNP96YOUhzyIDdSkffcTsONxSHUPn/ADm2HJ9e7vvT+Hb14Lv/9vbPrWA/xB4DHem/4iX3jfW1j+fbeO70eu4z2ZH/DkDT9n7cVLT1nOrz29i4UzSthxpJcvP/HqCeP//ZNvYeWc4YEzmfH4yD8/z4cuncOff+floeGRUIA/WrOAf/jJzlOuc6T7P3Ix11xQf8ppXtrbyYWzyrDWBYGRHZiMle9b/uqx7Tzwsz0njPvBn72d8niYyniYL36vmUde2g/ApfOruO+DK4dq8XYc6eVvfvgKv9zTkfeDuovCQf6/t8+nJBZifk0JF84q4//++iAvvu4eAv6Za87ngnPKKI+HiYaCp1yW71v2tCd49XAvT24/yqONBymKBFlSX0bzwW56UjkumlXOhbPKWTG7nPk1JTQd6GZve4KeZJaiSCbJZHAAACAASURBVJDW3gzP726noSrO/Jo4b1tYy7uXzaS8KMy+9n52tvYyozTG4vrSYSf7Wc8nHAyw82gf39t6gOd3d3C4J8WyhnIOdiUJGBeodrclyOR8ggHDpfOraO/LDH0/jhcJBiiOBunszw4bPquiiL50jlDA8JaFNbz13GrOqytlWUP5KTvB6UvnePC513ly+xF6UzmSWY/9nckxv0+lsRAXz62kP+3RncziWTsUZgMGfOuCbHVxFN+6ZpAueLqwHA0F2dvef0KN8kgV8TBd/VkumVfJpfOr6UpmONiV4mc720jnfIojQWLhIFnPPyH4AlTGw6xdUkd1SYRdRxOEAoZg0FAVjzCnKk5FPEx1SQTPh/a+9FDQLY2FWTG7nEgwyJHeFK+3JWhPZMjkfFbOqWDpzDJi4SAdiQwHu5JEw0EW15cSCwfJeT7GmKH7V0XOBmstnm9J5XzSWW/g+xCiL5MjaAzGMPS9K42FyOR8+jMetaVRoqEAvekc2ZyPb93F0rKiEPHI1GhU1JvKsqs1weHuFGDJ+RbfuuNgPBKkuiRCbypHe1+Gjv4M3f0ZgoEAvrVUxMOUF4UpibptsRY6+zMc6k4RDhrikRDF0SA9yRyZnE9VcWTgwo37zvanPSqLI1QUhaktjVJfHqO8KEwkGMAY6M945HxLaTQ0rEXFeHQnswyes7b1ZehOZimJhohHgvjWcqg7xbYD3fQPvH+d/e7iVFEkRGksRG1plNJoCGMMiXSOrOdTFAlSXxYjHArg+ZaSaIhw0GAtxKNuvvhAq46jvemhY28kFGBmedEZbc94WGuxlqF1jWyZks55WAu5gW2QqUMhrEAGQ1jihi5WDQthW06Y9i9D3+Tm0I/h/f8CS99DTyrLsrt+NDT+7hsv4CPjaPbX3Z9l+V+6+V/7wzjhzesA+FTmk2QJ8Wc3XcH5q9aOPvNd5ScOC4Thg9+Gzb875jIA/G76Tv7542uo2PEIybqVLPlOMXVlMVbPr+Y/Xj54ynn33HMdJpuEUBQw7mxzlN4bXc3Dsd4mrbWksj4/3n6EYKaHdz92qRvxhXYaD/Xxt4+7sPfsTlcjVBoL8de/cxHf33qA/9x+dMzb9vl3Leb9q2ZTFAkSDBjCwQC+b0nnfGLhADnf8uZ7nqStL8Mn1izgQFdy2LPeCmVedZzX2/tZVFfCNRfU8/i2w6RyHi0dSW5ccQ5XLanj2gvr2drSRVkszKK6krxD5FhYa8l6dqgGx1pbmPX5Hvg593/fEcgkoGsfVC+E2kWnLVPzwR6qSyLMLHdNFq3vAYas59Pbup+fv7KP7+yJ0pPKsbyhnNlVcWpLo1w4q5xza0tGX3AuDZ2vQ7ILckmonA+Vc90ZyUm22fMtrx7q5rXWBBXxCP3pHId7UvSmctSWRikvCtNQWcTh7hQ/+s0RfrW3k0DAEA0FKI6GWDm7gqXnlPHakT5SWY9wKMDe9sRQLZxvLRY40JkklfVYUFvMxXOrqCmJUFsapbYkSm1plEgoQPPBHp55rY2Wjn4CxvDC6x1DD4KvLY1y1ZIZLB4I033pHAbDvJo49eVF1BRHyHg+Wc/yk1eO8J/bj5LzfGaUxvCsJZXxSGRyFPoWznDQUBGP0JPMYoGGgbAZCwcojYXp7s/QUBnnuotmcl5dCcGAoTjiTo5fOdxDzrcEDNSURFkys4ziaIh0zqMzkaU0FqIjkSGZ9YiGAvSlc8yqKKIiHinsRhRAbmDfZzx3vHFNewPkPB8LY+4h92zrS+fY197P7rY+uvqz9KSy1JfFSGY9wgFXxqO9KXpSOcpi7oR3QU0xmYEaZYOhNBbCt3Cgqx/fh6JIkLJYmKJIkINdSTzf4ltLzrMksx7xSJB0zqc/4z6zvaksnf1ZjvSk2NnaRyKdIxYOEgkG6M94Q/N7vqU/47GovpRZFTGO9qR59UgvvaNceBiLwZr+kcPqSmPMKIsyI5ojlumkKGSIpls50JXicLCe8qo6upMeoVAIa6AyHqEk6j7DLR0Jcr6loaqYsliIhso4fqqXA70ee9qT+F6WMttDtmgG82eUEQ4E8KzbL939WVo6++nPeGRyPt1Jd7EpRpoI7u8yk8SzAaImQwCLwRIlS7XpoYoeSk0SnwAhcoTx6CFOsz+PFBHqTCdl9OMRoN50UEkvBuimmIO2mj6K6LdRgvgYLJWmj4O2mlZbToIicgTJmhDRoIFcmihZuimhIh6myOQIBQIEwhE6+jNYa4a+p8GAIRQwFIWDZDz3/TapLqK5HopJkSaMj2Gm6eCQrcIjQJQs5SQoNf0UkaHU9BMLeDQEusj5PlkboMfGabG1tNlyAvhkCBPCo5NSjtoKYrh9FDcpZpk26ugkbtJkCNFj44TwaaeMrA0SwOIRoC1Qg42UcE5RDgMEgwbjZwnFK0jmwMYqSOd8Wrvde5nIBSiKBAlYj1A4PPTd6erP0pvKUhoLk/F8+lI5jIFQwOBb1/maZy3xsAuc/QP7qrwoDLkMqf5eQngkiUK4CC+XpaK4iJmVxRRHg2Sy7sIfxpDzLNUlEcoiUJrrgJIZVEYsmWQP6XQaiiqZWVVGfXGA3o4jHE2HSKRy9PuGXKgUz/PIZNN0Jz2C2X5iJkXWg8qIR8xk6DMlpKM19GYgFglSXexCeTQUoCgcJBQMUBkPEw0HaO1Nk8r6VBZHKC9yFwIOdCbp7M9QPhDo3cXXLMmMT0ksRGU87H6XfDv0GxkKuLA/ozTKjCl2m4pCWIEMhrC+G7q4ZCCEPepdyvrsfz9h2nnmEE9Hb4VLPo5/2XoWfPk3w8a//L4k5Re9CyLHdQufaIfeQ1B/IfQehqLKgcAysMzPPcaFZjePRu9wA85/Ny+9ZSP/+ou9/K/3LT/5FeTB2q6bvw/fes+Z7YRRdNgSgvhcn/lrWmzdsHEGn3//xCpW1IXhma/AL76a/4re+0145KMnH1+1gB0rP8/Vj7l9Wk4fL8c+MTT6/ekv8LI9lzSFPwF79E/fRl1ZjL3tCS6eW8lnHmnk/wzUfAHUl8X46WeuJBQw/N43fsEvX+8YNv+fXHEudaVRPnLZHIJtr0D1eZDsgBe/Cbufgvlr4ML3QjAMra+CCcCMxVA5D3wfBk5+6D4AiVYXbGa9CXY/DdFSqL/InT2Ejzs4+R7s+SnsehIOvQzBCDRcAtaHbBLOv87Nl+qG0pmjhmY6drvgYi0U10J/OzT9H3jtCahd4p6dV7UAWneA9Vz5AyFo3wVYtx0YN+54dRfBxf8NYhXuO/DC/3brKG9w29+xy62veAYEQ9C+GxJH3bK8DEN3asarYcZS8LKurLEy6O+ATB+c8yYornFl2P+i2zYv4/bf8eI10N8G0TK3/DmXuQexp7rg6MD32gSgYTWUzXTNj7MpiJZA+WxX/vZdbtm+56apmAP1y9x+TRyFqnMhEh94X3xIdkL3PjeNCbj3xPrQsQdK64Y1cR7G9917kOqGo83YqnPpyYUomzEHE4lDLgNe2n0mTmPwRDY8oqayI5GhI+GuQmc9n9rSKKGAoawozMGuFE0HurHWXY29aFY55fEwWHhxbyf7OvoHmmwGmVtdTH8mx7YDPexu66O2JEo0HOBgV4p0zqMv7dGbzFJXFuOVwz0n1GaOJmBgZnkRnf0Z+jMewYAZteOfurIosytdjV40FKQnleXc2hJWzqlgQU0JlcVhisJBvIEgUBILkc76tPWl6Uxk6E3naBkIt8YYDnUlKYmF2N+Z5FB3kmzOMrMiRs63tPWmqSqOUFUcYUZpzNXoBlwQ+cXudnYd7cMYQybnD3s+ZDBgqCqO0JHIuGbsNcUsnFFCJBRkb3uCWCjIga4ksyqKWHpOGUURF0JSA1fEe5JZUlmPjOdTXhShujgy9J4d6k7RlcwSMJDzLOVFYYqjQapLohzoTLK7rY9YKIgxkEi7UDNY4zEY6k8nEgzk9bzLGro5x7ThY9hva+li9M9qeVGYGaVR5lYXU1UcJpX18TP91Ad6CJscQQMGS7ntoq29g1eTpcwuynJRaR9V4RzhAJTkOggbH5tJUJ5rI5Ttc9+zcJyAl8bmUmRDJdhICckcBHP9RE2OkvQR4ulWvGCMYK4f30LY6ydEjsAJXXcdkzRx2kIz3GfSepTZPopJAoYOU06/jVLs91JvOk+Yt9/E2cMs2qigy5TTHawkFI6yMHCIOu8IJV4nEeMRNj6x1NgvPp5tngkSPO747pkQORMm6rsWBB5BcoEIqWApMT9BMlhKMlhKPNdFLNdHLhDBWI+4f2Lz97GyJoCxk9MBVcIUkwoUUeb3ELYZPIKkgsUUeX2kAnG6TDnGBDCBAH3ROjoDVWSCRQTCMRKhclI2QpnXRXXuMKWZNoozrZRk20mHyjA2R9hLEPNHb43hEaQvUErQ5ojbBDkTJhkowRJwnz+/65Sf19GkiRAmO+b5soRoNdUc8cuJ2jRpwiRthDbK6bClRMgRNymyNkSWIBnClJgUMdL02SJ6iRPCI0aGiMnRaUvotXFCxiNhY6SIECZHsUkRI8uMN93ATTd9YFzbdLYphBXIT9YsZuZRQ+8NXawu7ud3637IS3tPPFgCVNLDr2N/PPR6UepBMrjanQ8H/5O/Dj/gRtx1XFPBf7gY2kc0vXvnX8Jln3QnrzC8VuvaL8Flf8y4WAtfmju8ieIX2iEQhNefgflryCS62bd/H0V1C5lZFnPn3m07YOPqMa0it/g9hBa/C7Y/Cq8+dvoZxqtynjtJHdnMcrzCcXjX3+LNfRsv/+Ab/GHzMjopI4iHx0A4GGF1bY4bq1r48N7bh4/41DZ3gv/yt2HZ+12ABnfS+/oz8K83jVqEzDVfJvLE/zyz7RiPcLELYqke8I87qS2b5YL/yDB0vEipC3Y9B1y4SbS5MDSaQNgtv6zBBZjyBjdPKAq9R6BiNpQONBkNx924aBmUnQNtO+BX34Lu4zq2MQEXEnMpV9b6i1x5MwnAuqAXjrtpSma4cOPnYP8LcOAlCMWg7gIXxiLF7kTryDYXWIJRmLHEfQdyaVh0rVtGKAYHf+3KESt36060ucAairkQWDXfvdeZhFtXV4ubLhgeqOk7fXAAXLnjNS7YZY87yS2udWVK9wyfPlrm9p+17nuQSUBxtdu33ihNFQNhKKmDvsNu22eucOUuqXPb177LBdJY+bHtipW7/ZXsdME5VuaGJTvhUKPbv3UXus9MUaULzNUL3bb4OVeOnkMu/PYedMG054D7DETL3HtaMcd9pmYsdbWOgaArX+srYIJQez74Hplcjpf2J2jpdPcXpgYC1qK6UkpiITzf53B3msYDXRzoTBKPBFlUV0prb5qKeITqkgjJjEdFPMyu1gQtHf3sbU/Q1Z8lnfMpLwqzq7Vv6N7F8RpsSloZD1NTEiUeDXGgM0lJNDhwb2B6qFlmaSyEN1DDvryhnGUNFfSlc8QjwaEmyumsR3/Goz2RoTIeoSgSYNuBHva0JUjnPOZWF+P5lroyF5qaD7oaQXDhzbeWsliY+ECz07beNH2ZHPGBq9Azy2PUlromr9ZCMpmkO+VxNJFjZlmM82YUU5E9QpYQkVCAKttFT6CchI1wcWWKc6M9nBPqpizbRrztZVI5QzjbTS4YwwZjFBWXEcz144WK8NMJugKVhGJxcqk+8D28VB9FycPEk4cI9beSi1XQXzqfWPcuoqnh97j6kVIyVYugqJJQugtSXQTTPZhwkfueWw/6jroLmF6WE/svPpWBY3wo6i6YhGPus55NuWHBiLtgk+kDL+curASj7jhVXOMuVsXK3QWcSPHA93Km+xwP/gZ0tbgLahh3waV34DuIcd/ZaJl73XcEcmlstBRTde5A+aybrqjCXfBp3wl9re6YOnixqKzBfV+La1zZAkH3XYoUu+9hUYVbfijmlhUIuuNTca075gweD0Mxd0Er0QaHm9xFo9KBi0q+5443JTPc9zbR5r7T6V5I97l9Zn0oqXWtGfrb3XAvC9mEK1cw5PZnotXty3iVK0824Y5f6V63L5Kd7jhYVOV+F7y0m65ijpsnHHdl87Lufeg74o6DwbAra1Gl25ZYudvW0nOO7cdkx7HyGeN+n4NhV6a+o8cuiofj7jerbJa7ODZ4DDYBVz4v4/72Pbe8TMLtZ3DDghE3vZd147P9rlzRMtfCItXjypfuceUfXFbPwYFWIX1unbnUwMc0COWzXHkGf99S3W54rNytO1rqXueSrjzBMGT63TYHwm6abNLtZ+u77S89xy2v76jbzkiJey+THeSyGVI2RLxiBoFMwpXRy7jfkHDcfT98z+2zSLF7D0KxgfOLbvdbZH1Xnu4D0HcEG4ljsxn8bD/0HsGkOjHBCCZajJ/LYr2s2+5IMYFoMTbZiUn3YQNhbCiGCYYxyXZ3gQSDGfFd902Yzjd/juqrPz2OY8DZpxBWIIMhrOf6bi4tSXDTjB/wq32jP5A5iMeu2EeGDfujzKd4wl/N67EPHRs4GMK8LNxdc/KVf/wn7sv2jxe71wuuhPc/ePIr4mfDr/4FGy2luWwND/3sVZqOpvnuqu2EfvS5sS/j2i/B0hvdj0OyE177EbzwT64m4Ko7XWBp3w3X/LU7EP36X6FzL7RudyfEVQvgz37tvvDWd//7WXjuH9yJ27Z/G76+FR+GKz4PL22CZ/5XQXfHWTVY+7HqY+69Tna4fdF9wO23GYvdNO273IH8+Kt857zJTROvgoq5gHUHzb6j7iTCBNwPZMVs92N90XvdD2wm4U6ao6XuAP7CP7sT9Hi1Cx7tr7lpas6HtlddLdTCte7Hys+5A3wg5EJM3VJ3MA1FT9mU76TSfbD3OVeOXApmX+pO8gdr06aDbMqdGGYSLuRhBvZTP3TucSc6/R3ux/HwNjestN6FXWPce9fyC7cfK+e797hmoZunfaf73xj3g19c407u4tXuB7qoAmYudz/q/R2uiWXfETc8UgJ7/sv9QPcecvPWLnb7ufewO8FKHldTa4InhvOSerfu3jE2xw3F3IlY9bkDJzFZt8zj5w+EIVzkThIGf1wDIbfPwNUOn/dOtz/jVe6kpr/DnbCFYu6kbcHlbrpwzM0XKR3+mcll3HFl8PgRrx4an8pk2XE0wc6jfSRSWXozrnldkddDNtVHIBBiTqyfykA/8eISasJZwv2HyAWLKQ/n8A++TCjT47YpHHcBsnaxex9LarFHmkmlM8RMFj8UB+sR7Njljltd+9w8JgBzLnXbneoZOFFMu+Gl9e44me51J0PFte7ksbgGm+rBJrvwEu0ESmrwK+YTjsbddznTB937saluTPtOaHsNuva6/WPtsZpaO3DyOPjejFX1QvfeRYrdMcDPDZxQl7pjU7T02MWBcJGbNhhxzXzLzhnYjjZ3TKld7C4QlA2cOHfvd2Vr3+lO7Ioq3HtaXOM+B4MnsMW17oJCbOBCTmiwxt8cO3HvbnGfmfLZ7jg4OC4YdesKnPpe2SnHWncsOb41jbyxZPrdcXkwUIpr7WF9tz+y/e74GAi573gg6I4/U+wcQSGsQJ5cs5hzjhp6bujm0uIE9771Be578rWTTj8sbAF9Nsaej7/KRf8899jAZb8Hjd8eX0HOuwY+/J3xzTMRulrgVw/CT788fPhHf+h+TJe9/+yXwVp3EtR72NWWjPwyZpMu+BXPcP8/e29+65mxFNb8Tzj3Svjyeaev8bjqLnjzn8LRZvejXz7LnVx843J4+6fhHQNNTH3PXTEcuL9pqJmhyETyB67MR+Ku9hTcFc3eIxCKuKvSMHDFNeKuLCfaXK3XYNPJcJE7Ma6cd6xWYGQY7+9wNZU9B1yT1WSHO+GoOc99Bzp2ux/YQAj2/RxafumWm00yFNRGC4kjRcvdyWrvyPtWB5rYltS7K7yRUigqd+G1qMqd+I9Wszgq407+q+a74NO1b2yzFVVCzSJ3spVJnNgaAo7VLJ+pUJEL8uVz3PtoAu5iQdVALeTgVfxg2IV5LztwxbzeHVO9rCtv1QIXhOLVx2oARETkBAphBfLk5Ys554jh7659M9/4wu34lQs42J3kbV96CoB//cNL2Xawmw9fOoeL7vrRCSEMcA9x3nuG3bXf2TX+moWJ5HvT86pN6w547M/h9//N1Rq1vuqu2tec507yfv5VuOH/H7iKehr+wEnhdNwPIlPVYK1qutfVakXiLoQFw+67uvNJV+MxeDU02eUCXH+bC1Tls114sHagGVur+4527XWhI93rglDFHBcqYwPNPmMVLiSV1B1rMhWvdk15Bpuf1i8b/n3P9LsmZD0H3PGkZpELkBhId7tgWLXAhZjj5+vvcOUrqnDHEWPcNvQddQEtVuZedx9wATfR6vZDUaULjomj0LbTbV+6z5VtsNlsSd2w+4xFROTsOlUIUz+WedhHHVSfSwBoqIwPDZ9bHedt5x1rUthriyg1I26YPF0Au+3gwBXbw/C1t8IHH4KXH4IXB+4hG2x/PpVN1+BRuwj+4FH3d8Vs9+947/3nsS9ruu4Dkals8NgXLYWRWSJeBcveN+FFOqlIHBpWAaP+9p5c/LgHgR9/HCmtc/8GzVji/q9ZOHz+aIkLdyIiMqUphOXBjKg9fOXua9nTlmB21bFA9u5lM7m88e+4IrCVeyNfP3Eh5bOHdzwwaLB9d2k9fGag04OGS46FsDUnPmhZRERERESmD91wMg52lL8AYuEgS2aWDRu28UNvooMyvuuvYV5qC1vqPzt8Yf+98djfRQNXPv/omdFXbIzrwOPOLnjzLXmXX0REREREJp9qwsZjoCXMWJ+P8NaF1fxsZzsA137k01B8m3tm1/5fug4X/ugZ2PUTeNunxrj+Kd4MUURERERETkshLC9jC2GDAQygqnjgAcFz3+z+Acxc5v6JiIiIiMhvDTVHzMPIe8JOZ9XcyrNUEhERERERmW4UwvIw8indJ3PHu13vVe9eNvNsFkdERERERKYRNUccDzP439hC2MffvoBL5lWxrKH8LBZKRERERESmE4WwcRiMXoFxtEZcPrvirJRFRERERESmJzVHzIs/2QUQEREREZFpSiEsD9csrePJfU/S0jvKw5ZFREREREROQc0Rx8EO3BN2wTll3PLUp4gEIrz0kZcmt1AiIiIiIjKtqCYsD9Z3zREzfmaSSyIiIiIiItONQth4DNSEMc7nhImIiIiIiAxSCMuDteqYQ0RERERE8qMQlgfP9ya7CCIiIiIiMk0phOXB83NDf1s1TRQRERERkXFQCMuD5x0LYbnjApmIiIiIiMjpKISNgzWu1ss/Lnhl/exkFUdERERERKYhhbBxGOwc8fh7wjKeuqkXEREREZGxUwgbh8G7v3z/WO+IelaYiIiIiIiMh0LYeAxUhfVleocGpb207gsTEREREZExUwjLw9N7fzL093XfvY6V/7JyEksjIiIiIiLTiUJYHvxRHtasZ4eJiIiIiMhYKITlwYwy7JXOVya8HCIiIiIiMv0ohOXBjPJ85q9u/erEF0RERERERKYdhbBxsANVYEsqF58w7nDi8ASXRkREREREpiOFsLycWBW2s2vnJJRDRERERESmG4WwcRi8F6y9v/2EcaN11iEiIiIiIjKSQlge2pNtk10EERERERGZpgoSwowx1xpjXjXG7DTGfO4U011ijPGMMe8txHon2uA9YaN1zCEiIiIiIjIWZxzCjDFBYCPwLmAp8EFjzNKTTPcl4IkzXedkGWyOOFoIW1C+YELLIiIiIiIi01MhasJWAzuttbuttRng28CNo0z3p8C/AUcLsM5JNdpzwqLB6ISXQ0REREREpp9ChLBZQMtxr/cPDBtijJkF/A7w9dMtzBjzCWPMi8aYF1tbWwtQvAIaTF+j1IRt79iOtWqnKCIiIiIip1aIEDZaxdDINPL3wGettd7pFmat/Ya1dpW1dlVtbW0BildIbrNG22CAI/1HJq4oIiIiIiIyLRUihO0HZh/3ugE4OGKaVcC3/x97dx4fx13Y///1mZk9tSvJkizJtnwlzmE7zukkpNACSWg4C7TlWwqUllKOFspR2v4otBzNt2n5/XpQWiChBUraQI+Uuzm4knKEhDiJY8exnTg+ZVvWLa32npnP74+R1pIt2ZYlr+zk/fRDD+/Ozs58Zvej3c/78/nMyBizF/hV4DPGmNfMw74XxEwX5vhh9w/rWxARERERETnnzEcIexi4wBiz2hgTB14PfHPyCtba1dbaVdbaVcCdwO9Za78+D/teEDONhN384M0A/NLXf4kNX9pANajWr1AiIiIiInJOmHMIs9b6wLuJrnq4HfhPa+02Y8w7jTHvnOv2zyYzXR3xLevfUrudq+TYM7IHgN/7/u/VqWQiIiIiInKu8OZjI9bau4C7jlk27UU4rLW/NR/7XBAzDIG1plprt7/69Fdrtx88/OCZLpGIiIiIiJxj5uWPNT/XHDsSlollarcnX6rezDhxUUREREREnqsUwk7DsSEs5sZqt//iob+o3X7BshfUq0giIiIiInKOUAg7DceOby1KLJp2vR8d/BG7hnad+QKJiIiIiMg5QyFsNsbT17EjYY2JRj7x85+Y9in/s+d/jlvWV+hjw5c28Fc/+6spyx849ACfevRTcyrizsGdbPjSBt7zg/fMaTsiIiIiInJmKITNwZ//3J8D0JXpoqOho7a8Pd1eu/3PW//5uOdd/1/XA3DH9jumLH/Hd9/BP239JzZ8aQNjlbFZl6caVvnVb/0qAPcduI//3Pmfs96GiIiIiIicWQphszAxAOaM33jtBa9l629upTXVyorsitp661vX851f+c4pbXPH4A5e+43XsrVv65Tl133lulmVbf/ofq781yunLLv5wZt58X++eFbbERERERGRM0sh7HRYePfl756yaHF6ce32Tw7+hCWZJUdXt5aDYwfZN7oPgMsXX1577HXfeh27hnfxhrvecNxuHu55eMYiBGHANXdcw4YvbeBbnQvhFgAAIABJREFUz3yLV3ztFbXH7v2Ve2tXbOwv9vPGu944ywMUEREREZEzZV7+Tthzhjn639svffuMq1XCypT7l95+ae123IlTCSusbVnL9sHtxz33hV0vZGXjSm5/8nZ++97fZlXjKv7jlf9BOpYG4I13vZEtfVumPOdDP/5Q7fab1r6JpZml/OD//IBr7rgGgC19W9jwpQ3c8oJb+FnPz/jtS36bTUc2YTB8/KcfB+DxNz+OY5TJRURERETONIWwWZiYjmgsGHN6fwNsIqBNF8AA/vGGf6ToF7n9ydsB2Du6l2u/fC23v+x2rmi/4rgANtnfvejvuHHljQCkvBRbf3Mrn938WT7z+GeAo2Ht67u+ftxzL7v9Mj5y3Ud43YWvO63jEhERERGRU6OhjzPgxcuj87D+4Ko/OOF6r13zWgCu7rx6yvKUlzrub4y9+e43s+FLG47bxkNveIj3XvlevvGab9QC2GS/e/nv8oWbvnBK5f7zn/45tz5+K72FXrpz3fTkewjC4JSeKyIiIiIip8ZYa0++1gLZuHGj3bRp00IXo+aeV17Iyl0uX7rB4a8+ve24x7+y4yvc8tAt3P3Ld9OV7aJQLXDtl6+dflu/cg+d6U7KQZn7DtzHB3/0QboyXdz9K3cDUPJLXH3H1dM+d1FiEUPlIe79lXtZmll6yuXvyffQEGvgy9u/zMtXv5y8n+filovZP7qf27bcxjef+eaMz3WNy2O/8dhpjwBOp+gXSXmpGR8fq4yRiWfmbX8is+WHPq5xCWyA52jigIiIiJw6Y8wj1tqN0z6mEHbqThbCrLXkqjka4421ZT35Hl5y50sA2PLmLbz1O2/l4Z6H2fqbU6+GuLl3M5ctvmxKyPm3J/+NpZmlrGleM+XCG9/71e9NuST+fPn4Tz/OnU/dOePjrclW7vs/9512EDtRKJ1wadulHM4fpq/YN2X5f73qv7i45WIAPvrARyn6Rf7yBX+J67inVZazWaFaIB1LU/JLdOe6SXgJujJdp/S6B2HAo72P8uTAk1y//HpGq6Osb13PWGWMlJfCGEOhWmDv6F4e73uczb2beaz3MZY0LGFl40rObz4fg+GCRRdwzZJriDmxOR+PtZafHPoJY9UxesZ6SHpJWlOt7B3ZSzWs4oc+u0d288zwMwCsaV7DTatv4sYVN2KtxRjD432P09nQScJNMFIeoS3VRlOiqbaPaljFjP/zrY+DA4YZy99b6CXtpUnH0lSCCnfvuZtqWKUn38NDPQ8RhiG7hndRCSt4xqMSVujKdLGyaSUvW/Uynr/s+bQmW6Pjw+p8ShERETmOQtg8ufuVF7Jql8vt1zv85WeOD2EzmXiNjTGUgzKFaoFFyUWz3v/EdMRjA9x8+8D9H+A7+77DD3/th1gsPzv8M/7oh39Ue/x/f+1/aUm2nPL2rLVTLk5yus5vOp9nRp6Z9rE/3PiH/NzSn+P85vPPygbxH9z/B3x333eB6OIpP9/181y35DqMMXziZ5/g37b/22ltd3l2OQfHDtKWbKO32Hva5Uu6SUpBacqylJeqTY1tTjSzKLmIFy9/MUszS3lm+Bnu3XsvD/c8TMkv8bylz+PGFTeSiqXY1LOJPSN7aIg1sKVvC8PlYfaO7j3h/pc0LGFF4wpCG7JjcAe5So6YE6MaVmd8zuqm1ZT8EkcKRwhtOO0661rXsaZ5DblKjv5iPykvxfaB7eSqOQwG17j41p/ynNZkK0kvybrWdTQlmij5JbLxLEfyR9gxuIND+UMANMYbGa2M4hqXxngjyxuXU6gWyMQyLMsuoz3dzrKGZXiOx6O9j9KWamNjx0au6riqdqGduQrCgG0D29gzsodyUCbmxGhKNJGNZ+kr9HHBogtY07ymFuArQYVqWKUh1jAv+xd5rir5JUIbkvSS037nWGvr3kET2pBCtcDBsYP4oU/BL1AJovPQ426cbDxLvppnpDzCvtF9DBQHSHpJEm4Cxzj4oU/SS9IQayBXyTFYGmSoNEQmnomO1U2Sq+aw1rIouQjXuIQ2xHM8KkGFdCzNsswyOhs6WZRYREOsgaSXxBhDsVrEYmlPt+MYhyAMCG1IzD31jr6SX8JiKftlfOszXBomG8/iOi6VoBKdSjHWzVBpCNe4lIISQ6UhKkGFhJugMdFIW6qNTCyDxZJwE3jGozHRSMJNUPSLtf8L1QKO4+Aal7SXJuWlyFfzGGMwxuDgkPJStKfbax2c9RSEAY5xMMYQ2pByUMZgaq9pyS+R9tK1juogDAhsQMyJTSlrNajO6j2YENoQay2u42KtrXV+hjbEMc6zsoP8dCmEzZO7X3Uhq552+dfrHT7y3q8S6+jAbW6u2/67c90kvSRtqba67XNCX6Gv9kemj/Vnz/szblp1EzEnxsu/+nLecdk7ePnql/Pg4QfZM7KHT2/+9JT1X3/R63nZ6pexvm09+0f3055uZ/vgdnYO7mTH4A4eOPQAX3/112lKNGEwfHnHl/mrn/3VlG2sb13PtoHpg/CXX/5lNizewEh5hLd9521sH9zO9cuv5z1Xvofh8jCVoEJHQwfnNZ03p9ekUC3wD4/9A3ftuYvB0mBteSaWIe7GCWzASHlkTvtIuAnKQfmU11/ZuJLOdCdXdFzBvtF9jFXGGCgNMFYZq01d9UOfbDzLmuY1XN15Nc9b8jyMMfQX+3mi/wme6H+Cgl9gc+9mtvbPLfBn41mWNCzhhV0vZGPnxtoU3MP5wzTGG1ndtJqCX2B5dnntOaENue/Affyo+0eUghJlv8xVHVdRCSsMl4dpTbYyUBxg59BOAhuwqnEVgQ1wcPAcD8c4OMYhX83z5MCT7B7ZDcBFiy6iFJRY3bSaFdkVjFXHKFQLtKXauLLjSlY3rcY1Lksalsz4hWqtZdORTTze9zh7RvaQ9tJYLCPlEbpz3cTdOL71OTB6gOHyMJbjP1/jTpyLWi5iTfMaHOPQPdZNwk2wLLOs1jgyGIp+kQ2LNzBUGqK/2E81rDJcHibmxLDWMlYdI1/NE9gTn7eZcBMszSylNdnKjsEdlIISV7VfRSaeYUnDEnYN7+Lg2EHiTpy4GyflpVjZuJIlDUvIV/Mcyh8itCGZWIalmaUYTC18NiWa8K1P2kszUBxgReMKruq4io50B0kvycGxg+wf3c9oZZS1LWtZ2bgSgMAGDBQHqIQVOtIdxN346VYxOUtNjGKfSNEvkq/maU404zke1lqKfrEWCBJe4rjRbD/0MRgsFs/xKFQLDJYGyVVyUXAoDzJaHqXoF0m6SQ7kDrCicQXLMss4nD/M9oHt9BX7yFfzDBQHKPiFWjiBaFQdCwkvQSaWqY2WJ9wEFks1qHKkcIT+Yj+BDUi60ch+JpYhsAFFvwjAUGmIclBmeXY5SxqWENqQhJdgcWpxLcAENuDpoacp+SU6GzpJekliToxDY4dwjEMpKFENq4Q2pDnRTFOiiaZ4E45xSHpJQhvy1NBT7BnZw2ApOu5jO5VOJObECGwwYydW3InTnGwmV8nVgtZEB85IeYTQhrUQMHE8J+MYpxZ0gNopCY5xauGuMd5IJahEr6+XxGAYLA1ypHDklI9tQsJNEHfilIPycVeunk9JN1k7Nt/6tCZb8RyPdCyNwdCd68ZzPMaqY2RiGQp+gcZ4I4sSiyj6RSphhUWJRbQkW2hMNBJzYoQ2jD4rSwMcyR9htDJKNaiSr+aj2R/GkPbSlIPytB2WrnFJeSnibpzRyih+eLRupL00gQ0oB2WysSxJL8lYdSy6H8/WQvRwebgWaP3Qr33nDZYGKfpF4k6c0IZT6p3BkI6l8UOfuBsn7aVpSjTVypOORffjTpyx6hglv0QmniHmxGiINZCv5in6RRpiDbQmW/FDn+HyMAW/QHOimcXpxbQmo985Y0zt+z8k5OJFF7Nm0Zoz9j6fDoWweXLXqy5k9dMu//ZihzfdFxJftYrz77l7oYtVN0OlIW7675tqH56z9cCvP0A2nj2t5x4cO8hb730rf/3Cv+aStktqy3OVHD8++GM+9sDHKPiF09r2hPdf9X7ecPEb+GH3D0l6Sa5sv5KiX2S0Msry7HK2D27nTXe96bS2fWX7ldz6klvZN7qP+w7cx2c2f2bK43e99i6KQZEV2RUkveQpbXO4NMzB/EHWt64/rTKdCmstm/s2M1AcIO7G+emhnzJYGiQTy3D1kqt5yYqX4DouB0YPcN+B+3CMw3lN53HtkmvpKfSwpGHJgo9MLkSPNFDrkR4uD3PZ4suohlU29WziJ4d+ws7Bnewc2okf+pzXdB5Fv8j+3H4ysQznN59POSiTcBPsGNxB3I1z4aILCcKAbDxL3I3Xvswy8QyrGldx2eLLSHpJSn4p+rKqFmhNtbJzaCfbB7ZzOH+YI4UjLM8sZ3F6MY8ceYTh8jADxQG6sl2sa13HWGWMwdIggQ04PHaYXDWHYxzakm21xsNoZRQAz0Tnx82mwQdRo2+iMTfRYPMcj850J9l4lmw8i2Mcegu9teA3Vh1jTfMaLm+/nLUta6f02oY2pOgX8RyP5dnlLG1YWvce6QnWWvqKfewb3UdzopkVjStIuImTPmd/bj/loEzSTWKxdDZ0kq/myVfyxN04uUoOgPaG9tpU94k6fTh/GM94tKZaGa2MYq2lKdF0wvMXy0GZg7mD+NanUC3gGpekl4waXslFjJZHowZ9ZbT2Xk008FzjcmjsEEPlIUbKI+wd3Ut3rpuYE2OkPIIxhpgTo+AXGCgO0JpspeAXyMQzJN3klPfcYDiQO1DbdnuqHYs9bip6NpalK9sFQPdYN4VqITpH00QN3FwlN21nxwTHOFNCRjaepT3VXqtvzYlmctUcaS8dBaXxUaFSUCJXyTFaGaXBa4hGGkw0et7Z0ElHuoN0LM1gaZDB0iD5Sp6QkMZ4IxZL2kvTGG9k7+he+gp9tZkwfYU+hspDteCysnElmViGI4UjlIMy5aAcdQRhaiHUNW7tPRkuDwPUjmlpw1IuWHQBi9OLaYo30ZhoZFlmGXEnTtJL1kJOOSiTq+RIeSmaE810ZbtoiDVgrSUkJAgDXMel5JdqU+Ib440n7JA69nWvBBW6c90cKRxhpDxC3s9T8qPG+0R468n3UApKZGIZHOPU3r/QhhgMQ+WhKPQZj5gbo+gXCWxAa7KVZZloZkHKS+EYh5ZkC7lKrnbObke6o/beTB7Vm1CoFhgoDVCoFmqvyUQDvxJWaoEm6UbPC2w0elSsFsn7+drfXw1tiMWSr+bpLfRS8kuU/FIthMfdeBTSw4CCH9XX5dnltQ7QifdhrDrGcGmYpJck7sQZKg8xXB5muDxMGB6ts23pNjrSHTQnmqOgEm8g4SZq2487cRoTR0+DqYZVUm6K0cooBb9ANaiSiWdIe+laEC1UC3iOR0OsgcHSYC2MTQS2odIQeT9PU7wJiyXmxEi4idpo4KLEIjLxDOWgjEPUKWCxuMalGlYZq4wRc2KUghL5ap7R8igh0UhtwS8wWh6tveYpL1V7H8eqY9EIqpuk4BcYLA7iOi7NiWZSXoqR8ghD5aEZf9/fc8V7eNulb5vx8YWgEDZPJkLYHS9yeOP90S/I2h3TX2r+2ezA6AEO5g+yIruCHx/8MTc/ePMJ17/tJbdxbee1dRme3je6j1d+7ZW1+1d1XMV7r3wvd++5m6/s+Mq87y/hJvjwtR/m6s6rWZxeTLFaZKg8xJHCER478hhvXv/mE079OpXeYnl2mpjCMdHTP11dmGhonakAeaL6N1oZJe2lpzToJxqJ2VgWY0xtpLcaVlmUWMTe0b081vsY/cV+in6RzoZOlmWW0ZpqZUvfFp4eehrXuHiOx7LMsijA5w5wYPQA5aDMSGWEIAxoS7VhiUb7srEsTw8/TW/h5NNtW5IttKZacXBqjXk/9GlNtZKNZzk8dpiEmyAVi6baNngNxNxYLdR2NHTQnmqPGhbGiUJBoY+YGyPmxMhVcrWwcXjsMEPlITzHo6/QV2tETPCMR0dDBy3JFtrT7QRhgDGGgdIAzYlmXOPycM/DjFXHTum9MhjaUm3EnBi9hV6MMTNO121LteGHfm02wUQjtb/QT66aO6X9nUzcibMks4TVjaup2irNiWhWSMkv0RBroCXZQn+xn8Z4Y6232zVuNJoUVrHW0pXtYmlmKUOlIXryPVgsK7IramGuFJQYKA6wf3Q/rhONUjfGG6PpbDYgX83TmmylPd1Oc7KZtJemJdlSC1j5ap62VBu9hV72je6rnft6NnzmWhsFj9l+L0602SbOgz2dqWQi57JqWGW4NFzroHJMNApmMLVTJ84mCmHzRCHs3NBb6KUn38N5TefNeHVFay0DpQEa4438b/f/0hRv4pOPfvKk0+8+/4uf55ol12CtpRSUTnh1RxGZP72FXnYM7qAclPGMh299POPVeqx3j+xmW/82BsuDVILoQiqBDaJe6UI//cV+urJdhDakElZq09istVTCSm2U4tje/clTthpiDWRiGaphleXZ5TQnmgltWLtQTEuyhRWNKyj5JXYO7eTQ2CH6Cn30F/troa41GY1alfwSl7dfzuWLLycbz1IKopByOH+4tp+CX6Ah1oDneHTnutk7upfQhnSkOwhswOqm1VSDKgOlgVowHi4P01foI+7GGS4PRyMCbpJyUKY52czi1GK6sl14jhdNoQujKUl5P09/oZ/GRCOtqVYaY42EhMScWO08Ej/0aU+316aQLvQot4jI2e5EIUzXXJ6N8c6zrvRSoHtBiyIza0+3055uP+E6xpjauXUvWRldvfLLr/jyKe/DGKMAJlJHJ/u9vm7pdXPeRzWsMlAcIOWlaqNaTfGm6LyDMJjVqMMreMXJVxIRkecsdWPNwiGzGIDXrnrVApdERETmW8yJ0dnQWRvVakm24DoujnE07UtEROaVQtgs2PGXy/gnvwKQiIiIiIjIdBTCZiU6V8D6s7simIiIiIiIyASFsFmw4+eEjd1//9FlZ/GFTURERERE5OyjEDYrUQqr7N59dFGgqYkiIiIiInLqFMLmyFan/zstIiIiIiIi01EImyOdHyYiIiIiIrOhEDYrx5//ZX2fYCyPrVQWoDwiIiIiInKuUQibo9y93+GpjRvZ8/rXL3RRRERERETkHKAQNgtmmgsh9nzsYwCUn9xO93veW98CiYiIiIjIOUchbBbMSR7Pfec7dSmHiIiIiIicuxTCZkV/E0xEREREROZGIWw2TiGDheXymS+HiIiIiIicsxTC5llYKCx0EURERERE5CymEDYLM50Ttug3fqN2O8zn61MYERERERE5JymEzYP4ypW122FeI2EiIiIiIjIzhbB54GQa6Lr1swAMfuHzC1waERERERE5mymEzcr0V+Zw4nEIo8dGvvHNehZIRERERETOMQphszDdH2sGMPE48dWrAHCyWbZfvJaem/9v3co1WZDLsf3itWy/eC02DBekDCIiIiIiMjOFsHlg4nESq1fjpNOEuRwAQ3fcMesQNHj77ex9/a/PqSzPvOQXa7cH/vnzhKXSnLYnIiIiIiLzSyFsFma6OqKJx4HjL08fFoqnvG1rLUdu+UuKmzcTjIzMumwH/+iP2X7xWoLh4dqyvr/9W3ZefgW9n/zkCfdrrf4ItYiIiIhIvSiEzcZM0xFdd9rlfm/vjJuqdHdTfPzx2v3cd75bu/3Utc8j/+BDJy3O8Ne/zug991J++mlGv/Wt2vLlt91KwwteULs/cOttVLq7Gf7a12tTFSd+dqxdx46169h+8VrGfvjDk+5TRERERETmxlvoAjwrmGiMzEmnp4yG7X75y1m7Y/txqx/5y79k8Eu3A9QeLzz88JR19v/Wb3H+975H9dBBGq65prY89/3v0/2ud89YlMXvey+ZF76Q5Pr15H/6U/I//gkj3/gGz9z4kpMexoG3v4PUlVey/DOfxm1uPun6p6P4+OPgOBR+9jDB8DCL3/feWoit9vbiNTfXRhZn4g8MEBYKxJcvPyNlfC6yQQDGYJz698v4/f2EpRJOQwO2VMJJpc5Y/TsZW6lQ6e7GOA6lnU/hNDSQuGANtlTCbWrCaWxckNdIREREnl0UwmbBTBoKa3vP75P/0Y8pPvZYLYRlb7qJka997aTbmQhgAKWdO+n9//6a/I9/HG33936P/s98BoBnbryxtt4FD/wEr6VlxgBmkkku3vxY7b7X1kbTq15F06texcg3vlFb3vKWtzB8550kzj+f4ubNrPnB98FxOPDO36W8YwfFRx/lqeddx7K//Rty992P39eHm83i9/XR+ZE/I7lu3am8VMcp797D/t/+bfyeninLB/7pn45bN7XxKoqbHqndd5ua6PrsZ0ldsp6hO+/kyJ/ffPS4Ewk6P/JnNLzg5/EWtz0rG8jWWow5OhnWhiF+Tw9eR0ctwNogoPTEE5Sf2U21uxtrQ9KXX47f14cNQopbHqf81NMkL74IJ91AccsWnFSSysGDVA90QxjiZDLEV6zAWovb0EDjK15O8pJLiK9cWetgcBsba+UIxsYYu+9+/P5+vLY2Yp0dBKOjVPYfoLL7GTIvvp7Stm04mQxjP/gB1cOHCQsFnHQaJ5MhzOej8k1z3mL8vPNoffvbcJubiS1ZSv5HP8T6PrGlS/H7B7CBT2LNGmy5TPnpXYSFAt7ixVGQjMcIRkZwM1myN1xPbNky/MFBqocOQ+BTPXSIyr59eJ2dEFr83iPkH/gp4dgY1Z4egsHBGd8LJ5ul8aUvJbH2YmypTDA8jJNOE+vqil6nZAKvsxMnnT763lQqVI8cwWtrwySTU95LEZGFYH0ff2AAiE6pcOLxqAPU82qfUTYICEZHcZJJwnyesFTGVso4qRReezs4zln3eRYWi9ggBL9KMDKCPzCAcV3cpibcRYuijrSzrMzy3GXO5vOBNm7caDdt2rTQxaj5l1dex7W7onOu1u7Yzt7X/zrFzZtZ+eU7SF95JdsvXjtl/fh553H+Xf9z3Hae+rnnz9jQW7tjO/7AAE8//wXHPbbqv+9k76/8au1+6vLLWfb3f8/ot79Fy2/91ozTIqtHeun/7Gdofctbpvxh6cnCcpnDH/5TRr/97ekPflz7H/8xmRe9iNjSJeA4VHbtwmtvx2trA6DwyCPEli3D6+ggHBujuGULB976Oyfc5nxbc/99xDo7sZUK3b//HmwY0v5Hf0issxOnoWHG12muJi7EYisVnGRyymPVgwepHDhAmM8TX30e8dWral8EwViesft+QHzlSka+/W2Gbv9XAJLr11Patg2AxEUXYWIxwlyOyr59te0mLriAyt692Gr1tMpsEgmafulVeB2dVA/sp9J9MApHhw9PPTfRcSAMiS1dionFCPJ5gv7+U96Pk8mQunQDTkMDpZ1PReGqo4NYVxdOKoWTToFxMLEYlQMHyN1zD35f32kd03T7DotFCIIZ14mtXEFi1WqchgZSG6/Cicfx2tvxBwcJR0dxGhsJR0Yobn2C3A9+gC2c/I+ym3gcjMGWy0fL0thI4oILcNJpvNZWsJbKvn2YWAxvSSe2WMJWqziZDP5AP8kLLwTAVn2cdIpgZBRbKRMWS2AMYaFA9fAh8AOcxkachjRuJoP1A8q7dpG++mriK1Zg4nGM51J8fAt+Xx/pjRtx0ilstUowNoZ/pDfabyoFnkvDtc8jufZicF2CoaHoPcpkcDIZCEOCoaHxMi8hGB7GzWSigHlMJ4gNAqqHe7DFAm5zM7ZSwW1pAcchHB3FbW7GxGKn+c6KRJ1UVKuEpVL0MzoK1mKDAFv1saUiJpWOOolyOSp79hKMjGCrFcLc2Hi4KGErFQiD6LPUdTFeDON5GM8Fz4uCSLFEWChgy2XCUhFCi5NpwEkksKHF+lXwA2wQRLNixju3onWSmGQCJ5EA18O4DmG5TNDfj7WW2JKl0Ynnvo+tVrHVKmGhSFgqYoslvCWdJM4/HyeVwvo+1g8g8AkLBfy+PoKx6FiC/gEqBw5E2yiVotcnCKLvpzCs3Z6u8wsAz8NraQHXjU6pOMHnJgDj36dOIhF9BiQTuA0Z3JYWnEwGE4thS9HrhutEv/OuF30Pey7G9cB1jltmK5XoGCZej1Ix+iypVGqPheUoEBKGhJUyYb5AeLJz6l0Xk0jgxGIQj+HE4rVyUz362kevsQ+OwcRik+pD9EPMw3ixqMMtET+6Tmz8x3Ww1Spua2v0WT0yGnUSeh4mPmm9RBJbLhGWyxgvhtuYxSST0XMzGfA8wrE8wcgIwcgw+EFUXwO/VtdspUJYKh7d5kQ54rFoG4ANbfT+23DKbYi+l6L3qYxJJDCJeK2++r192MDHlsrRaxGPY8uV8d+XMCpLuRx9x9R+ov3bYjTDxW1pwWttiX6P4nFMMoXb3ITT0EAwNATGwUklcVIpTDJJMDhIMDqK29iIu2hR9Hq7HiY2Xkdc97ggbe34MTHzKUILxRjziLV247SPKYSdumND2JFP/L8MfvGLnH/vPcRXruTgH3yA0bvumvKcix7fHH3oTrL7Va8itnQZY//7v1OWt77zHbS/730AlLZvZ89rf3nacix+//up7NlDx4c/hJvNztfhAVFF3rH2+NGu9MaNFOb4XsRWrOC8b3w9+tA8Zp/+kSN4HR0MfvFfKDz0EMv+7m+xvk9YLDHyta9S3Pw4Y/ffD0Qhy21qIv/gg1QPHCD3ve9T+NnP5lQ2gK5P/yOZ66+num8fbltb7cNrcjlHv/0/HP7Qh0hu2IDf00P10KE57TN1xRXRaOp0XJf0FVeM9066YCEcGyPIjeI1LyLI57HVCqlLNlDt6cFraSFzw/Uk167FXbSI0rYncRJxCo8+hruoGa+1lcT55+N1dGCDEDfTMGO5rLUUHn4Yv6eH8q5nCMdygKG8ZzfeohachjTe4nbS116Lt7gtCoK+jy0WSaxdS6y9ncLmzSQvuggbBMQ6OmbV2La+T+mJJ6j29uLt+AEXAAAgAElEQVT395N5/vPBGIKRUUzMw+vooPjYYxjPI7l+PU4qRZAbw5aKUVBwHKrdB8n94Pv4R3rx2tpIXLAGJ5PBbWomceGFVPbtBSCxahUmnT7l3tGwUCAYG8NJJjGJRBSEug9SevJJsGHUcMrno9fM83ASSdymRmy1Snn3Hip79xKMjuD39oHvk9ywgbBUxO/rw0mlCYaHscUisa6uWmOqNhK5qBknkax9mTsNDeA6eIsXY0tlwrEx/KGogye+rIvili2EY2NHq1RzM15nJ+UdO6Yck9fejkkkogv7TDQgT4Pb3Exs+XIIAvz+fvz+/toXY40xMP69Y2IxEmvXkly/Lgpo2UZMzMPvH8B4Ht6STghCvMVtJC/ZQHL9uqhRaEwU5PIFwlwU5o79XJFTM9F4qY3cWhvVwVLpaIO0Wo2+azyPyp69VLsPUD3cg997BL+vHzwXJ57AxOOEpRLByHD0nKZm3EwDJpWKzqm29mij2q9G4b9SiTol2loJcmNU9++PGlmxow1VpzELoR0PTEVsvkBQyEd1vlSC0+yEmmCSydrn09GQM94In/g/DMF1o0Z3PB59r3seYT4fBRrXPRoivNj44y7hWJ5wbIywXJ62nCadjl73aX7nTCoVNUwTiZMGIpNK4WazuC0txJd3YeKJqHPLcaPGs+NGI1eOEx1HMonb1opx3PFQEzWsw0L0WYS1eJ2deK0thPk8TraxduxhbhR/YBDCMHptjImCVqmELZUIcjmCwcHotfF9TCqJk0pj/SphbuxogAjD6PUNojBRu12tHg1KsfHwk4gT6+iMXo+JEBOPPn+jxxM4qSReRyfGdaL3KJPFW9wWtSVGR/GHhggGh44GvIkwN15uE/NqdQ4vCllRMPcn1YlqFNbG74f5fLQdf2qAo1qFWCwKGdbW2mqTf6emvoFHPxdnfI/HRyonZlpE9S0KhbV6OzlElkrR57/jHB21PPZ2GBLkclH9jsWm/10yBpNMRgF+/L0x8TjGGEwqFQWziddzUkg2ySS2WDzpcZ0Wz5sy22RiH4vf9z7a3vmO+d/fHCiEzZNjQ5j1fcq7d9d6qwHyDzxA8tJLeWrj1QC0vu1ttH/gDyjt3Bl9KDsOz/ziTSx6wxvI//SnVPbsAWDF7V8ivXHjlJ7k8jPP4GQy2HKZZ37xptryCzc9fFxAqIf+W2+j7x/+4eQ9Y8doeu1raXzlK6KG9BlWeOQR9r3xTbX7yfXr8RYvxu/rq40qAcRXrpwyonRKZviQjJ93HpXdu4mffz5e+2IKP33wuHViS5fS/icfZORrX6e0bRvx1aspPHh0vZbffDPJSy8ledFFJNasIRgZOdrIlucsG4a1kVUbhrOebjvRoDaOQ5AbI7ZsKcaYaGQQoh7GSdOPIPpCKz39dBTUHBevrY2wWKgFHqzFbW6ORkMHh3AbGwnLJWwxarxVD/eAMXgd7XgtrXhtrZhEEsIgGuncfyBqIKXTVHsOU3z8cap790VXah3vvfcWLyYsl4/v1R4vp0mlouPI52sPOek0yUsvJb5iRdRZkR/DzTaONxqiBkO1tzcKxc3NONksXmtr1PBKJHEyDXitbcSWLcNJJaMe2aFhwtGRqCe7WCIcy0XnMOYL0TFXKthy1AsdjowQ5PNU9x/AJJPE2ttxslmcTANutpEwP4aTThOM5qJed6C0ZSuVffuihmosFjVsYrHaCKqTTEYNxDAgvnQZsWVLCcuV6LFMphaOnIYGwnIJ/CA6d7EhHTWox8tnq1WM60bTcvfvwx8eJswXoobgeKMrarRG04Qnj97OyHGiHu62tqhxVqkQVis4sThuSwvG8whGRggLhaOh3nHGe+qjhqPX2opJJqPRnIF+nIYG4stXRMfvV2sNu2BkBONGDU0Ti+Fks7jZbNS4T6aiRmkygZNM4mSzR3vLPQ8nkYjel+6DuE1NxFetwmttiY43m42eewq/V+FEb/8cprLZIMCWy7WGstPQEDVuIRrBM+ZoCDimtz8sFqkePBi9lxMjA140quO1tkbL5KwzMUPmuFkC4yOUYT4fdQLE4xAEBLnceCCMReG9UolGhBqjEHzGyhiGGM+rfeeExSK2WIym+U+eonrM6REn3XYQEAwN4Q8O1j5vwmKRYGSUMJ/HXbQIsFHnZbGALZZwW1twm5oJhoYIRkfGO0LGR/0mjTZbP/rschIJMA64Dg3XXkv6qqvOxMt02hTC5sn3rrucZUPRl9N0F9yYbOeVVxEWCjS/7nUkLrqII/936h9v7vjwh2n+5ddSePQxEhesIdbZecLtdb///eTuvofkunWs/up/z+1A5miil2MiINggwLhu1COVSlF87DGS69fXGhP1nn9d2r6d4uOP0/y61x03LG2tBWunfCCWdu4kGBrm8Ec/QnXf/tpyp6npuEZg+ppraHzlK4ivWIHb1ETi4otnPL7y7j3EVyw/4ZdjkMtF0zhOcjESkeeKibBpw5Awl8NJp/F7e8k/+CCVvVHHSZgfA9cj1tGOk22MvuR7j1Dc/DjVw4drjetgeLjWuAkrZWLtHYTlUm3a0umO+NU4TtQDn0jgNDbiZjJ4S5eAH1A92E1YKkdTxEZGoqA0NoaTzUY9t0DiwgtIXngRJpWc1JNcjTq6XBfrV2shofzMboLBQUwiAa5DMDCI29hIkMtF51qmUtHo4Phx1ULd+LRYjMFraSHW1YXX1oqTbjga+lwHWywS5KKgGFsyfl7jpFGBYHg4mpK8bBmJ887DbW09btq1iIhMpRA2Tyaf83WyEOYPDfH0dT834+MT5y3NRvGJbXiLFxPraJ/V82TuJs47UFgSeXaw1kajD240HSscHaV6+DB+/0A05a1Uxm3M4mQbwYY4DRmcbAavrS26nUyc8ujDRO/x5P+BM9ZBdTqjpiIiMv9OFMI0fn0aum797EnXOdm5Wl777INU6pL1s36OzA9jDCiAiTxrGGNwm5qOLmhpIb5q1Rnb13T/nykKYCIiZz+FsNOQfdGLTrqO8Ty8zs4pl2Rvfec7aLzpJkpPbteXpIiIiIjIc5RC2Bk0OYC1/+EHaP2d6FLtybVrZ3qKiIiIiIg8yymEnUGx5cupHjhA1z/+A9lJf3hZRERERESeuxTCzqDVX/sq1YMHSV500UIXRUREREREzhIKYWeQm8ngKoCJiIiIiMgkujqEiIiIiIhIHSmEiYiIiIiI1JFCmIiIiIiISB0phImIiIiIiNSRQpiIiIiIiEgdKYSJiIiIiIjUkUKYiIiIiIhIHSmEiYiIiIiI1NG8hDBjzEuNMTuNMbuMMR+c5vE3GmO2jP88YIy5bD72KyIiIiIicq6ZcwgzxrjAp4GXAeuAXzfGrDtmtT3AC621lwI3A5+b635FRERERETORfMxEnYNsMtau9taWwH+HXj15BWstQ9Ya4fG7z4IdM3DfkVERERERM458xHClgEHJt3vHl82k7cCd8/0oDHm7caYTcaYTX19ffNQPBERERERkbPHfIQwM80yO+2KxryYKIT9PzNtzFr7OWvtRmvtxsWLF89D8URERERERM4e3jxsoxtYPul+F3Do2JWMMZcC/wy8zFo7MA/7FREREREROefMx0jYw8AFxpjVxpg48Hrgm5NXMMasAL4K/Ia19ql52KeIiIiIiMg5ac4jYdZa3xjzbuBewAW+YK3dZox55/jjtwIfAVqBzxhjAHxr7ca57ltERERERORcMx/TEbHW3gXcdcyyWyfd/h3gd+ZjXyIiIiIiIueyefljzSIiIiIiInJqFMJERERERETqSCFMRERERESkjhTCRERERERE6kghTEREREREpI4UwkREREREROpIIUxERERERKSOFMJERERERETqSCFMRERERESkjhTCRERERERE6kghTEREREREpI4UwkREREREROpIIUxERERERKSOFMJERERERETqSCFMRERERESkjhTCRERERERE6kghTEREREREpI4UwkREREREROpIIUxERERERKSOFMJERERERETqSCFMRERERESkjhTCRERERERE6kghTEREREREpI4UwkREREREROpIIUxERERERKSOFMJm6YnzGxa6CCIiIiIicg5TCJuFwQaPocbYQhdDRERERETOYQphs2CsBcxCF0NERERERM5hCmEiIiIiIiJ1pBAmIiIiIiJSRwphIiIiIiIidaQQNgs6G0xEREREROZKIWyWrJKYiIiIiIjMgUKYiIiIiIhIHSmEiYiIiIiI1JFCmIiIiIiISB0phM2aTgoTEREREZHTpxAmIiIiIiJSRwphIiIiIiIidaQQNluajSgiIiIiInOgEDYLxi50CURERERE5FynEDZrGgoTEREREZHTpxAmIiIiIiJSRwphIiIiIiIidaQQJiIiIiIiUkcKYbOgs8FERERERGSuFMJmySqJiYiIiIjIHCiEzYquUS8iIiIiInOjEDZrGgoTEREREZHTpxAmIiIiIiJSRwphs6HZiCIiIiIiMkcKYSIiIiIiInWkECYiIiIiIlJHCmEiIiIiIiJ1pBA2C7ouooiIiIiIzJVC2GwpiYmIiIiIyBwohImIiIiIiNSRQpiIiIiIiEgdKYTNmuYjioiIiIjI6VMImwWjP9YsIiIiIiJzNC8hzBjzUmPMTmPMLmPMB6d53BhjPjX++BZjzJXzsd+FoBwmIiIiIiJzMecQZoxxgU8DLwPWAb9ujFl3zGovAy4Y/3k78Nm57ldERERERORcNB8jYdcAu6y1u621FeDfgVcfs86rgdtt5EGg2RizZB72XV8mXOgSiIiIiIjIOW4+Qtgy4MCk+93jy2a7DgDGmLcbYzYZYzb19fXNQ/Hmj+8Y4m7jQhdDRERERETOYd48bGO6ywUee+rUqawTLbT2c8DnADZu3HhWnYJ13aNPct1CF0JERERERM5p8zES1g0sn3S/Czh0GuuIiIiIiIg8681HCHsYuMAYs9oYEwdeD3zzmHW+Cbx5/CqJzwNGrLWH52HfIiIiIiIi55Q5T0e01vrGmHcD9wIu8AVr7TZjzDvHH78VuAt4ObALKABvmet+RUREREREzkXzcU4Y1tq7iILW5GW3TrptgXfNx75ERERERETOZfPyx5pFRERERETk1CiEiYiIiIiI1JFCmIiIiIiISB0phImIiIiIiNSRQpiIiIiIiEgdKYSJiIiIiIjUkUKYiIiIiIhIHSmEiYiIiIiI1JFCmIiIiIiISB0phImIiIiIiNSRQpiIiIiIiEgdKYSJiIiIiIjUkbHWLnQZZmSM6QP2LXQ5jtEG9C90IeRZRXVK5pvqlMw31SmZb6pTMt/Oxjq10lq7eLoHzuoQdjYyxmyy1m5c6HLIs4fqlMw31SmZb6pTMt9Up2S+nWt1StMRRURERERE6kghTEREREREpI4UwmbvcwtdAHnWUZ2S+aY6JfNNdUrmm+qUzLdzqk7pnDAREREREZE60kiYiIiIiIhIHSmEnSJjzEuNMTuNMbuMMR9c6PLI2csYs9wYc58xZrsxZpsx5r3jy1uMMd81xjw9/v+iSc/5k/G6tdMYc9Ok5VcZY7aOP/YpY4xZiGOShWeMcY0xjxljvj1+X/VJ5sQY02yMudMYs2P88+o61SuZC2PM+8e/954wxnzFGJNUnZLZMMZ8wRjTa4x5YtKyeatDxpiEMeY/xpc/ZIxZVc/jm0wh7BQYY1zg08DLgHXArxtj1i1sqeQs5gMfsNauBZ4HvGu8vnwQ+L619gLg++P3GX/s9cB64KXAZ8brHMBngbcDF4z/vLSeByJnlfcC2yfdV32Sufp74B5r7cXAZUT1S/VKTosxZhnwHmCjtfYSwCWqM6pTMhv/wvHv93zWobcCQ9baNcDfAZ84Y0dyEgphp+YaYJe1dre1tgL8O/DqBS6TnKWstYettY+O384RNWyWEdWZL42v9iXgNeO3Xw38u7W2bK3dA+wCrjHGLAEarbU/tdHJm7dPeo48hxhjuoBXAP88abHqk5w2Y0wj8AvA5wGstRVr7TCqVzI3HpAyxnhAGjiE6pTMgrX2h8DgMYvnsw5N3tadwA0LNdKqEHZqlgEHJt3vHl8mckLjw9xXAA8BHdbawxAFNaB9fLWZ6tey8dvHLpfnnk8CfwyEk5apPslcnAf0AV8cn+b6z8aYBlSv5DRZaw8Cfw3sBw4DI9ba76A6JXM3n3Wo9hxrrQ+MAK1nrOQnoBB2aqZLyLqspJyQMSYD/DfwPmvt6IlWnWaZPcFyeQ4xxrwS6LXWPnKqT5lmmeqTHMsDrgQ+a629AsgzPsVnBqpXckLj5+m8GlgNLAUajDFvOtFTplmmOiWzcTp16KypXwphp6YbWD7pfhfRELvItIwxMaIAdoe19qvji4+MD5Ez/n/v+PKZ6lf3+O1jl8tzy/OBXzLG7CWaCn29MebfUH2SuekGuq21D43fv5MolKleyem6Edhjre2z1laBrwI/h+qUzN181qHac8anzTZx/PTHulAIOzUPAxcYY1YbY+JEJwF+c4HLJGep8bnFnwe2W2v/dtJD3wR+c/z2bwLfmLT89eNX7FlNdALpz8aH3HPGmOeNb/PNk54jzxHW2j+x1nZZa1cRffb8wFr7JlSfZA6stT3AAWPMReOLbgCeRPVKTt9+4HnGmPR4XbiB6Jxo1SmZq/msQ5O39atE36kLMhLmLcROzzXWWt8Y827gXqKr/XzBWrttgYslZ6/nA78BbDXGbB5f9iHgr4D/NMa8lejL6nUA1tptxpj/JGoA+cC7rLXB+PN+l+hKQSng7vEfEVB9krn7feCO8c7F3cBbiDpnVa9k1qy1Dxlj7gQeJaojjwGfAzKoTskpMsZ8BXgR0GaM6QY+yvx+330e+FdjzC6iEbDX1+GwpmUWKPyJiIiIiIg8J2k6ooiIiIiISB0phImIiIiIiNSRQpiIiIiIiEgdKYSJiIiIiIjUkUKYiIiIiIhIHSmEiYiIiIiI1JFCmIiIiIiISB0phImIiIiIiNSRQpiIiIiIiEgdKYSJiIiIiIjUkUKYiIiIiIhIHXkLXYATaWtrs6tWrVroYoiIiIiIiMzKI4880m+tXTzdY2d1CFu1ahWbNm1a6GKIiIiIiIjMijFm30yPaTqiiIiIiIhIHSmEiYiIiIiI1JFCmIiIiIiISB2d1eeEnW123bOLTGeGzss7F7ooIiIiInIOqlardHd3UyqVFrooMk+SySRdXV3EYrFTfo5C2Czc8bI7APio/egCl0REREREzkXd3d1ks1lWrVqFMWahiyNzZK1lYGCA7u5uVq9efcrP03REEREREZE6KZVKtLa2KoA9SxhjaG1tnfXIpkKYiIiIiEgdKYA9u5zO+6kQJiIiIiIiUkcKYSIiIiIiInWkECYiIiIi8hy0detWOjs7eeKJJ87qbZ6LZTgZhTARERERkeegW265hQceeIBbbrnlrN7muViGkzHW2oUuw4w2btxoN23atNDFqPm4+TigS9SLiIiIyOnZvn07a9euXehiyDyb7n01xjxird043foaCRMREREREakjhTARERERkeeY2267jc7OTi6//HLOO+88/uVf/gWAYrHIC1/4QoIgqK377ne/m5UrV9buVyoVfuEXfgHf92e9vdtuu40lS5Zw+eWXs2bNGl7zmtdQqVSm3eaZPKbpjutMl28yhTARERERkeeYLVu28LGPfYzNmzdz55138oEPfACAL3zhC/zyL/8yrusCsGfPHu6//34qlQq5XA6AeDzODTfcwH/8x3/MentbtmzhlltuYfPmzTz11FM88cQTbNmyZdptnqljmum4znT5JlMIExERERF5jtm6dWvtHKaurq7aKNEdd9zBq1/96tp6H/3oR/nTP/1T1q1bx7Zt22rLX/Oa13DHHXfMentbt27liiuuAGDXrl1Ya7nwwgun3eaZOqYTHdeZLN9k8xLCjDEvNcbsNMbsMsZ8cJrH32iM2TL+84Ax5rL52K+IiIiIiMze1q1bufjii7HW8qlPfYpXvvKVVCoVdu/ezapVqwDYtm0bTzzxBL/2a7/G2rVrp4SVSy65hIcffnhW25vY5pvf/GbWrl3LVVddxRe/+EUaGxun3eaZOKaTHdeZLN9k3lw3YIxxgU8DLwG6gYeNMd+01j45abU9wAuttUPGmJcBnwOuneu+RURERETOVfe87x56NvfM6zY7L+/kpZ986QnXOXDgAGNjY9x0003EYjGuueYaPv3pT9Pf309zc3NtvQ9/+MPcfPPNGGNYu3btlL+75bou8XicXC7H8PDwKW3vwIEDtLe3s2XLFgBuv/12br75Zr773e8et81sNlt73o033khPz/Gv01/8xV/URrhO9ZhOdFynW77TMecQBlwD7LLW7gYwxvw78GqgFsKstQ9MWv9BoGse9isiIiIiIrO0ZcsWbrjhBu65554py8MwpFQqAfDQQw9x7733snnzZt71rndRKpW49NJLp6xfLpdJJpOntL2J/a5bt652/7LLLuNv/uZvpt3mZN/73vfm5ZhOdlynW77TMR8hbBlwYNL9bk48yvVW4O6ZHjTGvB14O8CKFSvmoXgiIiIiImefk41YnSlbt27lssuOPzto0aJFBEFAqVTiQx/60P/f3t0HSVLXeR7/fCvrufphnnpmmCcYdXjcG2CYRREVEXSB8wTPlcBbhN1VCTdE3TsNxbuL5QzDh43d8Nb1VBYRRVdBVtFlDcQH1nXlQpERFRiRc0SEYZ56emb6uR4y83d//CpzamZ6hu6umupueL8iOrrqV1mZv8z85sP398vM0re+9S1ddNFFkqTdu3en90pJ0tDQkAYGBpTL5aY1vmKxeMg9W8453Xbbbbr44ounHOfxmKdisXjM+Tqe9TtcJ+4JsynKpvwFaDO7UD4Je//RRuacu9k5t9k5t3lgYKAD1QMAAACQeOSRR47o1Uq85jWv0Y9+9CPVarU0UZGkFStWaHx8XPv27ZMk/eAHP9Bll102rfHdf//96XBf+MIXdPbZZ+ucc85RtVrVhz70oXTY1nF2ep7uv/9+fe973zvmfB3P+h2uEz1h2yWtbXm/RtKOwwcys42SbpF0qXNuqAPTBQAAADBDx3rC3/XXX6+Pf/zjaeLUanh4OH39la98RR/96EenPb6LL774WZ8s2DrOmZpOHb70pS/p1a9+9RGfJ/N1POt3uE70hD0oaYOZrTezvKSrJN3dOoCZrZN0l6Q3O+f+XwemCQAAAKDDzj77bF144YVH/LBxq3q9riuuuEKnnHJKR8Y303HO1HTrcCydrp85N+WVgzMbidllkv5OUiDpVufch83s7ZLknLvJzG6R9AZJv29+JXTObX628W7evNlt2bKl7fp1ygftg5KkG92Nc1wTAAAALESPPfZYet8RnjumWq9m9rOj5TyduBxRzrl7JN1zWNlNLa/fKumtnZgWAAAAACxkHfmxZgAAAADA9JCEAQAAAEAXkYQBAAAAXdSJZzJg/pjN+iQJAwAAALqkWCxqaGiIROw5wjmnoaEhFYvFGX2vIw/mAAAAAPDs1qxZo+3bt2twcHCuq4IOKRaLWrNmzYy+QxIGAAAAdEkul9P69evnuhqYY1yOCAAAAABdRBIGAAAAAF1EEgYAAAAAXUQSBgAAAABdRBIGAAAAAF1EEgYAAAAAXUQSBgAAAABdRBIGAAAAAF1EEgYAAAAAXUQSBgAAAABdRBIGAAAAAF1EEgYAAAAAXUQSBgAAAABdRBIGAAAAAF1EEgYAAAAAXUQSBgAAAABd1JEkzMwuMbPHzWybmd0wxeenmtmPzaxmZu/txDQBAAAAYCHKtjsCMwskfUrSqyVtl/Sgmd3tnPtVy2D7JL1L0hXtTg8AAAAAFrJO9ISdK2mbc+4J51xd0h2SLm8dwDm3xzn3oKRGB6YHAAAAAAtWJ5Kw1ZKebnm/vVk2K2Z2nZltMbMtg4ODbVcOAAAAAOaTTiRhNkWZm+3InHM3O+c2O+c2DwwMtFEtAAAAAJh/OpGEbZe0tuX9Gkk7OjBeAAAAAHjO6UQS9qCkDWa23szykq6SdHcHxgsAAAAAzzltPx3RORea2fWSviMpkHSrc26rmb29+flNZrZS0hZJfZJiM/tLSac750banT4AAAAALCRtJ2GS5Jy7R9I9h5Xd1PJ6l/xligAAAADwvNaRH2sGAAAAAEwPSRgAAAAAdBFJGAAAAAB0EUkYAAAAAHQRSRgAAAAAdBFJGAAAAAB0EUkYAAAAAHQRSRgAAAAAdBFJGAAAAAB0EUkYAAAAAHQRSRgAAAAAdBFJGAAAAAB0EUkYAAAAAHQRSRgAAAAAdBFJGAAAAAB0EUkYAAAAAHQRSRgAAAAAdBFJGAAAAAB0EUkYAAAAAHQRSRgAAAAAdBFJGAAAAAB0UUeSMDO7xMweN7NtZnbDFJ+bmf198/OHzWxTJ6YLAAAAAAtN20mYmQWSPiXpUkmnS3qTmZ1+2GCXStrQ/LtO0mfanS4AAAAALESd6Ak7V9I259wTzrm6pDskXX7YMJdL+qLzfiJpkZmd0IFpAwAAAMCC0okkbLWkp1veb2+WzXQYSZKZXWdmW8xsy+DgYAeqBwAAAADzRyeSMJuizM1iGF/o3M3Ouc3Ouc0DAwNtV66TTn/j6Vp26rK5rgYAAACABawTSdh2SWtb3q+RtGMWw8x7ZibnpswdAQAAAGBaOpGEPShpg5mtN7O8pKsk3X3YMHdLuqb5lMSXSBp2zu3swLS7a6r+PAAAAACYgWy7I3DOhWZ2vaTvSAok3eqc22pmb29+fpOkeyRdJmmbpAlJf9budOcMHWEAAAAA2tB2EiZJzrl75BOt1rKbWl47Se/oxLTmkhldYQAAAADa05Efa34+4Z4wAAAAAO0gCZsJE5cjAgAAAGgLSdgMcDkiAAAAgHaRhM0QlyMCAAAAaAdJ2AzEYaz9v90/19UAAAAAsICRhCUpkNUAABc+SURBVM3A1ju3SpKe/OGTc1sRAAAAAAsWSdgsTAxOzHUVAAAAACxQJGGzwfM5AAAAAMwSSdgs1Mfqc10FAAAAAAsUSdgs3PeB++a6CgAAAAAWKJKwWaiP0hMGAAAAYHZIwmaDe8IAAAAAzBJJ2CzEjXiuqwAAAABggSIJm4GlJy+VJPWs7JnjmgAAAABYqEjCZmDVH66SJFmG6xEBAAAAzA5J2AxkAr+49j+xf45rAgAAAGChIgmbCTrAAAAAALSJJGwGuAwRAAAAQLuyc12BheSFf/RC/eLzv5Ak7dm6R8vPWD7HNZp7Y7vG9NAtDylXzkkmnXPdObKMac8je3TgyQM68PsDOuvas1RZXpFzTuFkqEw2oyAfzGg6LnYa3zOuyvKK6mN1VQ9UVegvKKyGihuxxnaNKVvMKqpHGt0xKuec6mN1FRcVVRmoqLioqPHBcS1ev1jlZWU99o3HtP+J/SovLSushupZ2aNcOaegEChbyCpbzMoypqAQKKpHKi0uKSgEKi0uaXTnqLLFrOqjdfWt6fPz/hwWNSJlgoyiRqTqgaomhyaVq+RUXlaWmWls15hqozU988AzCvKBgnyg3tW9qgxUVOgvKFvIqrS0JLOpGzGcc0f9bKFzzimqRcoW/a42akQaenxIoztHVRuuyTKmsBoqakSSpCAf+OXWV1BtpKb6eF2FvoJ6T+hVaWlJLnIKa6HCyVD5nryCfKDSkpIy2WO3pznnVBupyUVOljFVh6tykS+TpHxPXpP7JxXkffzne/MqLSkpV8odMg4zUxzFsoyl68w5J0np+7AWamJwQhN7JxTWQsn5+RrdMaqh3wyl21J5aVn53rxy5Zxy5ZyiWqTGZEOZIKNCX0G5Sk5BLlAmm9HE0ISyxawyQUZhNfTbaj7Q+J5xSVIcxelTa3OVnOpjdWWyGZUWlxSHscb3jCuOYgX5QC52KvQW/DoxSc7PQ5APZBnzyygwZYtZxWGsbMH/jxqRXOzS6WRyGWULWQWFQHEYK8j52JdJteGawprfN03un1RUjxTVIr98S1lVD1TlYqd8T175nrxfbtVQUT1K61dcXFRU99teJptRVI8UR7Ey2YxypZyyRT/t5DL51vWUaEw0VB+rS06K6lF6NUcmyCishX79mjS+ezwdf7qOzQ4ZV5D36yIOY4XVULXhmoJCoCAXHFKXTDYjC0xx6JdTobeQ7u9d7ORip6gRKW7Ecs4pX8n7snrk/xqR6qN13+hpSsuDXKBcJad8Ja+wFqo+VldjvJHup5N6ZLKZtD5hNUyPC2bm11kx69dbc19lGVNjsqHGREOZbEaZIOO323qkOIxVH6unwybLNN+TV7G/6OPJOdVH6wqr/tgW1kK5yEnml3N1uKqoFskyfrnWx+qKG34bihp+35Ar59L4yWQzPoaa26vMb1sudsqWssr35NM4lclvP6Wc3yYDS+PZMubH5fxyt8CPo3XbnWr7PZ7iMFZYC/12ncuk00z2KXEjTvdJYS308RI1Y7C5HJLlGhSCgzHq/PaYyWYUN+L0tYucX6fNZdq6z4pq0cFjj/n4PmK5uEP3LXEUS0l1mss7iZfGRMMv3+a6Cquh36/kAllg6XQsY+k6iKM4nZdkG43DWNUDVVnG0njO5Pw2n8lljlhXcRQrDuN0e0rjukPr0zmnOIzl4ua8ZPz8pXHZxfg5XOuxKD3WFDo3791grTvZ+Wbz5s1uy5Ytc12N1MTeCf3NwN9Ikla/eLXe+pO3SvJB+unTP63zbzhfZ1171lxW8bgJq6FqIzV9/b98Xb+773cdGecJm05QJpvRyrNXqjZc05M/fFJrz1ur8kBZQT7Qqs2rVB+va3THqB7+0sMa/v1wR6bbSUE+UGVFRUE+UHFRUfWxuqJ6pEUnLdKi9Ys0cNqA+k/slyT1rupVVPMnGY2Jhqr7fSJZXlpWZXnFn6zV/AmHBX5HPfT4kLKlrGrDNcVhrNEdoxrfM67JoUkV+gppojmxd0L1cX/iku/Jq7ysrMZ4QzKpeqCquBErqvskKjkIlpb6k/feVb2qjdQ0MTjhhxmupid/QSHQgScP+J3uVLuK5knssykuKsrFTsXFReXKOY3vHle2lJWLnKrDVfWt6dPSk5f6JPqZUQX5wM/HQDk9iNaGa+mJ9MjTI4fMWzJMctCKw1ilJSVlS/4EbGznmD+QOOcPZtlMOv3aSM2fmJV8Yp3J+RPeQl9BPSt7VB4o+5PwWiQ5vy2M7hzV8FPDisNYhb5CevJUXlqWBeYTkMlQE0MTmhicUL7Xn2g2JhrTWl4zYRlTZXlFpSWl9EQrOVEO8oFqwzVNDE0cPJmZgWR5yqRwMpQF5pdDy0lLMp1Cf0GZIKOxXWOdncEFJj1J6ZJs6WDiETf8eg+rYdemv9C1s766va47JZPLKF/xyX9jsuGTxMAU5IM0wU8S2Tj0J/mNiYY/NjVPelsbAJJlENV9YhNW/QmxTGmCIDvy532SRHs2+6bZSKd3lONZksQGhUBxI04bEuad6Rx3TcoWsv6Y0ExGk/OK9Kqu5jpykT9mJEmh1GwwSZLVlhhPxnf466ROFlh6PE4S0kwukzZ4pNtMc9rJ9JJlnTQ4NSZ9A0syzmwhmzZcSFJjvJHu51rrceGHLtQr/ucr2lq8nWZmP3PObZ7yM5Kw6WtNwiTpRnejJOnpHz+tW196qyTpVR95lV7+gZfPSf2ezeiOUf3ooz/S1ju26oyrztCKjStU6C3o0dsf1eN3P54OV1xU1AU3XqDNf7FZu36+S/e++14989Nnjjre9a9arzUvXaNdP9+l+mhd+5/Yr/WvWq84jPWCV79AT//4ae3/7X7VRmpavH6xwqpvxR98bFD7frPPt9RKvmclMN+6fVhYlgfKOvOaM1Ubran3hF5lS1l/8pcLlCvnVFleSTfcnpU9/iS+2WI6MTih2mhNxf6ihp8a1sj2Ea172TqdeMGJaYvrxN6JtNU4rIaKI9/yUx+r+xarMFZjsqG9j+3Vkg1L0tam/b/dr5GnR9SYbCicDA+W/86XVw9UO74ey8vKKi4q+tb3fKCxXWPqW9uX9s6lPQTNHVzv6l7J+d6OTM4nClE90uT+yYO9iKWsysvKylfyyvfmlS1l08St/8T+tPU935tXaXFJYTXU+OC4olqkvjV9KvQVtGLjCt9zWIs0sn1E44Pjqo3U1JhoaN+2fQpygar7q6qP19Wzsic9Wawsr2j0mVHt/fVeBYVAi05clLZ0TwxOKI58AlRcVEwTpJ4V/mciSktLaU9FHMUKJ8O09TvpickEPtG0wBTkAn/SUY/SXtlCXyFdf7lKLv1fH61rbNeYJvZO+N7bZmtlsqyWvHCJMrlM2goeR7Em9vpkpzxQ9surJ69F6xel4yj0FbRkwxL1r+tXsb+oOIrTntekh2B8j19uxUU+Ya0eqGps15gmhybTk+1sMav6eF1RzQ8/umNU1f3Vgy3dGfO9S/VIuZ6cKgMVn3Q3W20LvQVlshnle/MyM9XH6yr2FxU1fI9NbaSmyX2TPsltJq/ZQjZNYuPIx4aLnHIV34tVHfbD9q/tV2VFRZWBStrbFNUilZeVtezUZaqN1BRWfYJaH6srnAxVH/dxmyvnfGI8WvM9Bs0TuNKSUtpTky1m1Rj367A8UE7nN8j79VMfryvfk5eLnCb3T8oypp4VPenBXaZ0ukmPQWtLanKSEFabSWfSINHs5Up6RJIemmS7jxqRj4NmA0CS7BcXF5UtZJXJ+Vb6xmRDpcUlWcZUG62pMd7wy7fZQ5OUT+6bTHvu4kac9s4n+6Jk+6mP1g/pYQgKfjkm23yukvMnzS1XHySNGWHV9zZUllfSnkbnXPq/tUU5aaVPYrDQV0jjJayFimpR2iqfnPRJUm20dsjJd9JDk7RYJ41Hyb4z2U6SXrOkPG74RCCJlUKv7y11sUsbr5Lln9QnW8ymjQTOuXT5Jz2TSY9cruy31WQ/kiyvTDajfE8+XdeS73mqj9VVHa6mDROFvsIhPafJyXwcxv5qgGI2ja1cJef3WZHvPQ2roRoTjbTeceh7Wwp9hXQ8fsH5hpD6WP2Qnq1wMlRjsnGw16il5yxq+B7ApGcyE2T8d2qhb6ST35/lSjm/LpvxnMRWVIsONnCVs2k8Jj2XLjrYkyQp3UayxWzaS5Jc+eIip1zZJ3hJ3CQn+sl4M9mMiv1FX69mvFtwMGFIkoE4ig+egDenn+wrku0sDuO09zHpyUmTh4wpW8oe7BlLemKbyyvZlpI6JD1QmSCTzquL/HJI4iVbOrj9WMbSOEiWU9Kz5tyh9Uj2Na1JU7G/mCZASYwmcdvaU50c49OkJ2NprCb76GS9J7GdJEDJ9p1s78k4kt6+ZLm3XhmUxEiQa+4DG1G6nSe9gK37gKTHLA59XZKexdZkLFkWyXjiMPZXJDWPick4k3hJ1kW+kj+4zYZx2kO87uXrtO78ddM7meqS45aEmdkSSV+VdJKkJyVd6Zw74tGBZnarpNdK2uOc+4Ppjn+hJGFP3f+UPv/yz6flfxX/VUe6Q5OD8OT+SfWv9b0pSWIw9JshLT15qQq9Be3btk+f3PBJveiSF+nKu65MN/oDvz+gmzfdLEm69t+u1W2vvK3tOl3++ct11p/63r5kA2x3Xke2j2hy/6RW/IcVkvxlCJK0/Sfb09b2VZtXPeslV/ONc04j20fSnoGxXWNpa1CunEuTmZFnRjSxd8KfuDUvS0l2oP3r+hXkgvSyvuQyLAAAAMxvx0rC2r0n7AZJ9znnPmZmNzTfv3+K4b4g6f9I+mKb05tTR0sCHv7Hhw95P7pjVH2r+6Y93p0P7dTN5/hk6YrbrtCZ15yp3Y/s1k0bb5pR/bbdu00fKX9kys9aE7Ar77pSD332IQ0/NazBrYNadNIi/cm9f6LaSE2rzlmlbfdu0wOfeEC//e5vJUmnvO4UveGONxxyf4iktDWkXX1r+tS35uDyyhZ8WJ50wUkdGf9cMTP1r+1PE+ijOWHTCV2qEQAAAOaDdpOwyyW9svn6Nkn/pimSMOfcv5vZSW1Oa84VFxWnLP/ZP/xMkrTpuk166OaHtPuXu2eUhD361UfT19+89pv65rXfnHHdNl69UWvOW6N73nHPIeXnvvNcrThzhf7lrf+iiz52kV72/pdJkk57/WlHHdeGyzZow2UbZlwHAAAAAM+u3SRshXNupyQ553aaWduPCzSz6yRdJ0nr1s2v6zoPt2/bwfuZJOnCD17ok7CHdx81iakeqOr2192u8rKyfv2NX6fly05dpk1v26Tvvue7aVlxcVHv3f1eSdIjX35E337nt/Wene+R5K/1j6NYg78a1MBpA2kv3carN6b3ebTa9JZNnZlpAAAAAG151iTMzL4vaeUUH/2PzldHcs7dLOlmyd8Tdjym0Smf3PBJSdLSk5dq5Vkr1bPSPyzgvg/cp/Pfd376BJpnfvqMHvz0gzr5P52sf/rjf5pyXH1r+nTefztPm962SZ846RO64H9doBe/88Xp52f96VnpvViJTJBJ76NKFPoKHZs/AAAAAJ33rEmYc+7io31mZrvN7IRmL9gJkvZ0tHYLxOiOUb3wkhceUvbIVx7Rxqs3KqyFuuXFt0iSfnnbL4/47imXn6LBXw3qituukOR/U+V9Q+87/pUGAAAAMCfavRzxbknXSvpY8/8/t12jBag+Vk8fmX3lXVfqzv98p77x5m9o49Ub9eHih48Y/jUff43OeOMZqiyv8KQ7AAAA4Hmm3STsY5LuNLO3SHpK0hslycxWSbrFOXdZ8/3t8g/wWGZm2yXd6Jz7XJvTnhNve/Bt+uwffvaI8sryiiTp1CtOTcue/OGT6es//79/rlw5p97VvaoMVI57PQEAAADMT2398JJzbsg5d5FzbkPz/75m+Y4kAWu+f5Nz7gTnXM45t2ahJmCStGrzKp33nvOOKC8vK0vyjyV/7T+8VtLBx8L3r+vX2peu1cqzVpKAAQAAAM9zC+vXb+eJi//6yNvkWi8rPOnCkw757NofXHucawQAAABgoSAJm4VMcORiW/KiJYe8Pvdd56bvF79gcVfqBQAAAGD+a/eeMDQtPXlp+trMdOknLtUrb3ylaqO1OawVAAAAgPmGJOw4Ki0pqbSkNNfVAAAAADCPkITN0pV3Xanvv+/7uuZfr1FxUXGuqwMAAABggSAJm6XTXn+aTnv9aXNdDQAAAAALDA/mAAAAAIAuIgkDAAAAgC4iCQMAAACALiIJAwAAAIAuMufcXNfhqMxsUNLv57oeh1kmae9cVwLPKcQUOo2YQqcRU+g0YgqdNh9j6kTn3MBUH8zrJGw+MrMtzrnNc10PPHcQU+g0YgqdRkyh04gpdNpCiykuRwQAAACALiIJAwAAAIAuIgmbuZvnugJ4ziGm0GnEFDqNmEKnEVPotAUVU9wTBgAAAABdRE8YAAAAAHQRSRgAAAAAdBFJ2DSZ2SVm9riZbTOzG+a6Ppi/zGytmf3AzB4zs61m9u5m+RIz+56Z/ab5f3HLdz7QjK3HzeyPWsrPMbNHmp/9vZnZXMwT5p6ZBWb2czP7VvM98YS2mNkiM/uamf26ub86j7hCO8zsvzaPe4+a2e1mViSmMBNmdquZ7TGzR1vKOhZDZlYws682yx8ws5O6OX+tSMKmwcwCSZ+SdKmk0yW9ycxOn9taYR4LJb3HOXeapJdIekczXm6QdJ9zboOk+5rv1fzsKklnSLpE0qebMSdJn5F0naQNzb9LujkjmFfeLemxlvfEE9r1CUn3OudOlXSmfHwRV5gVM1st6V2SNjvn/kBSIB8zxBRm4gs6cn13MobeImm/c+5Fkv63pL8+bnPyLEjCpudcSducc0845+qS7pB0+RzXCfOUc26nc+6h5utR+ROb1fIxc1tzsNskXdF8fbmkO5xzNefc7yRtk3SumZ0gqc8592Pnn6DzxZbv4HnEzNZI+o+SbmkpJp4wa2bWJ+kVkj4nSc65unPugIgrtCcrqWRmWUllSTtETGEGnHP/LmnfYcWdjKHWcX1N0kVz1dNKEjY9qyU93fJ+e7MMOKZmN/fZkh6QtMI5t1PyiZqk5c3BjhZfq5uvDy/H88/fSXqfpLiljHhCO14gaVDS55uXud5iZhURV5gl59wzkv5W0lOSdkoads59V8QU2tfJGEq/45wLJQ1LWnrcan4MJGHTM1WGzLP9cUxm1iPp65L+0jk3cqxBpyhzxyjH84iZvVbSHufcz6b7lSnKiCccLitpk6TPOOfOljSu5iU+R0Fc4Zia9+lcLmm9pFWSKmZ29bG+MkUZMYWZmE0MzZv4Igmbnu2S1ra8XyPfxQ5Mycxy8gnYl51zdzWLdze7yNX8v6dZfrT42t58fXg5nl/Ol/Q6M3tS/lLoV5nZP4p4Qnu2S9runHug+f5r8kkZcYXZuljS75xzg865hqS7JL1UxBTa18kYSr/TvGy2X0de/tgVJGHT86CkDWa23szy8jcB3j3HdcI81by2+HOSHnPOfbzlo7slXdt8fa2kf24pv6r5xJ718jeQ/rTZ5T5qZi9pjvOalu/gecI59wHn3Brn3Eny+55/dc5dLeIJbXDO7ZL0tJmd0iy6SNKvRFxh9p6S9BIzKzdj4SL5e6KJKbSrkzHUOq4/lj+mzklPWHYuJrrQOOdCM7te0nfkn/Zzq3Nu6xxXC/PX+ZLeLOkRM/tFs+y/S/qYpDvN7C3yB6s3SpJzbquZ3Sl/AhRKeodzLmp+7y/knxRUkvTt5h8gEU9o3zslfbnZuPiEpD+Tb5wlrjBjzrkHzOxrkh6Sj5GfS7pZUo+IKUyTmd0u6ZWSlpnZdkk3qrPHu89J+pKZbZPvAbuqC7M1JZuj5A8AAAAAnpe4HBEAAAAAuogkDAAAAAC6iCQMAAAAALqIJAwAAAAAuogkDAAAAAC6iCQMAAAAALqIJAwAAAAAuuj/A477/0hNXSbFAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 864x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "nn = np.arange(1, B + 1)\n",
    "\n",
    "\n",
    "f, (a0, a1) = plt.subplots(2, 1, gridspec_kw={'height_ratios': [3, 1]}, figsize=(12, 8))\n",
    "\n",
    "a0.plot(nn, np.cumsum(A_hat) / nn, label=r'$\\hat{P}(A)$')\n",
    "a0.plot(nn, np.cumsum(B_hat) / nn, label=r'$\\hat{P}(B)$')\n",
    "a0.plot(nn, np.cumsum(AB_hat) / nn, label=r'$\\hat{P}(AB)$')\n",
    "a0.plot(nn, np.cumsum(A_hat) * np.cumsum(B_hat) / (nn * nn), label=r'$\\hat{P}(A) \\hat{P}(B)$')\n",
    "a0.legend(loc='upper right')\n",
    "\n",
    "a1.plot(nn, np.cumsum(A_hat) * np.cumsum(B_hat) / (nn * nn) - np.cumsum(AB_hat) / nn, \n",
    "         label=r'$\\hat{P}(A) \\hat{P}(B)- \\hat{P}(AB)$', color='purple')\n",
    "a1.legend(loc='upper right')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As noted, the estimates converges to the theoretical value -- and the product of the estimates only converge to the estimate of the joint event in the scenario where the events are independent."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}