Project_14_Toto.ipynb

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   "cell_type": "markdown",
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    "# Odds of Winning the TOTO Prize\n",
    "---\n",
    "\n",
    "## Introduction:\n",
    "\n",
    "This project aims to help TOTO lottery addicts better estimate their chances of winning. \n",
    "\n",
    "The project will build functions that enable users to answer questions like:\n",
    "- *What is the probability of winning the Jackpot prize with a single ticket?*\n",
    "- *What is the probability of winning the Jackpot prize if we play 50 different tickets (or any other number)?*\n",
    "- *What is the probability of having at least five (or four, or three) winning numbers on a single ticket?*\n",
    "\n",
    "The scenario we're following throughout this project is fictional — the main purpose is to practice applying the concepts learned in a setting that simulates a real-world scenario. Throughout the project, we'll need to calculate repeatedly probabilities and combinations.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Core Functions\n",
    "\n",
    "We will build the core functions required to be used in our programs to calculate factorial and combinations.\n",
    "\n",
    "Before that, let's recap on the following concepts.\n",
    "\n",
    "\n",
    "#### Permutation\n",
    "\n",
    "Permutation is referring to certain arrangement where the order of the individual elements are important.\n",
    "\n",
    "For example, a 3-digit PIN code have 10x10x10=1000 permutations. This means we can have, let's say PIN code `377`, which is not the same as `773` or `737`.\n",
    "\n",
    "If we do sampling without replacement (i.e. no repetition of elements), the number of permutations or different arrangements that we can get from a set of 5 items are 5x4x3x2x1=120. This is represented as `5!` (meaning 5 factorial).\n",
    "\n",
    "However, if we were asked to choose 2 out of 5 items, then we will have 5x4=20 permutations.\n",
    "\n",
    "This is represented as <sub>n</sub>P<sub>k</sub> in the formula below, where in this case: n=5, k=2.\n",
    "\n",
    "![Permutation](Permutation.png)\n",
    "\n",
    "\n",
    "#### Combination\n",
    "\n",
    "Combination is referring to arrangements where the order doesn't matter.\n",
    "\n",
    "This is represented as <sub>n</sub>C<sub>k</sub> in the formula below, which we’re reading as “n choose k”.\n",
    "\n",
    "![Combination](Combination.png)\n",
    "\n",
    "In this example, “5 choose 2”, or <sub>5</sub>C<sub>2</sub> will have 10 combinations.\n",
    "\n",
    "\n",
    "#### Probability of Events\n",
    "\n",
    "Take the example of rolling a die, where the set of all possible outcomes, or the sample space of a die roll, Ω = {1,2,3,4,5,6}.\n",
    "\n",
    "In this example, let's consider the event E as \"*getting an even number when rolling a die*\". Thus an outcome for event E is considered as *successful outcome* if it is 2, 4, or 6.\n",
    "\n",
    "The probability of event E, denoted by *P(E)*, is\n",
    "\n",
    "![Probability](Probability.png)\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
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    "### Function to calculate factorials (to find the total number of permutations for sample without replacement)\n",
    "\n",
    "def factorial(n):\n",
    "    product = 1\n",
    "    for i in range(n, 0, -1):\n",
    "        product = product * i\n",
    "    return product      "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "### Function to return the number of combinations when we're taking only k objects from a group of n objects.\n",
    "\n",
    "def combinations(n, k):\n",
    "    numerator = factorial(n) \n",
    "    denominator = factorial(k) * factorial(n-k)\n",
    "    return numerator/denominator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One-Ticket Probability of Winning the Jackpot Prize\n",
    "\n",
    "We will write a function that calculates the probability of winning the TOTO Jackpot prize.\n",
    "\n",
    "In the TOTO lottery, six Winning Numbers and one Additional Number are drawn from 49 numbers that range from 1 to 49. A player wins the Jackpot prize if the six numbers on their tickets match all the six Winning Numbers drawn. If a player has a ticket with the numbers {13, 22, 24, 27, 42, 44}, he only wins the Jackpot prize if the Winning Numbers drawn are {13, 22, 24, 27, 42, 44}.\n",
    "\n",
    "We want players to be able to know the probability of winning the Jackpot prize with the various numbers they play on a single ticket (for each ticket a player chooses six numbers out of 49). So, we'll start by building a function that calculates the probability of winning the Jackpot prize for any given ticket.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "### The function takes in a list of six unique numbers and prints the probability of winning.\n",
    "\n",
    "def one_ticket_probability(list_of_6_numbers):\n",
    "    \n",
    "    ### Start by calculating the total number of possible outcomes — this is total number of combinations \n",
    "    ###    for a six-number lottery ticket. \n",
    "    ### There are 49 possible numbers, and six numbers are sampled without replacement.\n",
    "    \n",
    "    n_combinations = combinations(49, 6)\n",
    "    \n",
    "    \n",
    "    ### Probability of one successful outcome (i.e. 1 ticket).\n",
    "    \n",
    "    p_one_ticket = 1 / n_combinations\n",
    "    \n",
    "    \n",
    "    ### Convert the probability to percentage.\n",
    "    \n",
    "    percent = p_one_ticket * 100\n",
    "    \n",
    "    \n",
    "    ### Print the result in a friendly way.\n",
    "    \n",
    "    template = \"\"\"Your chances to win the Jackpot prize with the numbers {input} are {num1:.7f}%.\n",
    "In other words, you have a 1 in {num2:,} chances to win.\"\"\"\n",
    "    \n",
    "    output = template.format(input=list_of_6_numbers, num1=percent, num2=int(n_combinations))\n",
    "    \n",
    "    print(output)\n",
    "    "
   ]
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   "cell_type": "code",
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Your chances to win the Jackpot prize with the numbers [3, 49, 7, 2, 6, 9] are 0.0000072%.\n",
      "In other words, you have a 1 in 13,983,816 chances to win.\n"
     ]
    }
   ],
   "source": [
    "### Testing the function with sample input.\n",
    "\n",
    "input_1 = [3, 49, 7, 2, 6, 9]\n",
    "one_ticket_probability(input_1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Your chances to win the Jackpot prize with the numbers [3, 8, 7, 45, 6, 35] are 0.0000072%.\n",
      "In other words, you have a 1 in 13,983,816 chances to win.\n"
     ]
    }
   ],
   "source": [
    "### Testing the function with another sample input.\n",
    "\n",
    "input_2 = [3, 8, 7, 45, 6, 35]\n",
    "one_ticket_probability(input_2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Historical Data Check for TOTO Lottery\n",
    "\n",
    "We will also build a function to allow users to compare their ticket against the historical TOTO lottery data and determine whether they would have ever won by now.\n",
    "\n",
    "Below we will be exploring the historical data coming from the TOTO lottery for over a thousand drawings (each row shows data for a single drawing), dating from 3 Jul 2008. For each drawing, we can find the six Winning Numbers drawn in the following six columns:\n",
    "- Winning Number 1\n",
    "- 2\n",
    "- 3\n",
    "- 4\n",
    "- 5\n",
    "- 6\n",
    "\n",
    "\n",
    "The data set can be found on [Lottolyzer](https://en.lottolyzer.com/history/singapore/toto/) and it has the following structure as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
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    {
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       "(1296, 32)"
      ]
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     "execution_count": 6,
     "metadata": {},
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   "source": [
    "import pandas as pd\n",
    "pd.options.display.max_columns = 50 ### to avoid truncated output. \n",
    "pd.options.display.max_rows = 20\n",
    "\n",
    "url=\"https://en.lottolyzer.com/history-export-csv/singapore/toto/ToTo.csv\"\n",
    "data = pd.read_csv(url)\n",
    "\n",
    "### Get the number of rows and columns.\n",
    "\n",
    "data.shape\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Draw</th>\n",
       "      <th>Date</th>\n",
       "      <th>Winning Number 1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
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       "      <th>41-50</th>\n",
       "      <th>Division 1 Winners</th>\n",
       "      <th>Division 1 Prize</th>\n",
       "      <th>Division 2 Winners</th>\n",
       "      <th>Division 2 Prize</th>\n",
       "      <th>Division 3 Winners</th>\n",
       "      <th>Division 3 Prize</th>\n",
       "      <th>Division 4 Winners</th>\n",
       "      <th>Division 4 Prize</th>\n",
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       "      <th>Division 6 Winners</th>\n",
       "      <th>Division 6 Prize</th>\n",
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       "      <td>2021-02-18</td>\n",
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       "      <td>6</td>\n",
       "      <td>15</td>\n",
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       "      <td>6.0</td>\n",
       "      <td>74237.0</td>\n",
       "      <td>217.0</td>\n",
       "      <td>1412.0</td>\n",
       "      <td>428.0</td>\n",
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       "      <td>2021-02-15</td>\n",
       "      <td>9</td>\n",
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       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>87530.0</td>\n",
       "      <td>126.0</td>\n",
       "      <td>1911.0</td>\n",
       "      <td>380.0</td>\n",
       "      <td>346.0</td>\n",
       "      <td>7015.0</td>\n",
       "      <td>50.0</td>\n",
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       "      <td>2</td>\n",
       "      <td>3634</td>\n",
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       "      <td>2</td>\n",
       "      <td>2.0</td>\n",
       "      <td>774896.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>326272.0</td>\n",
       "      <td>157.0</td>\n",
       "      <td>1429.0</td>\n",
       "      <td>271.0</td>\n",
       "      <td>452.0</td>\n",
       "      <td>6691.0</td>\n",
       "      <td>50.0</td>\n",
       "      <td>8285.0</td>\n",
       "      <td>25.0</td>\n",
       "      <td>120531.0</td>\n",
       "      <td>10.0</td>\n",
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       "   Draw        Date  Winning Number 1   2   3   4   5   6  Additional Number   \\\n",
       "0  3636  2021-02-18                 1   4   6  15  40  44                  37   \n",
       "1  3635  2021-02-15                 9  13  18  20  34  40                  36   \n",
       "2  3634  2021-02-11                10  23  25  28  46  49                  36   \n",
       "\n",
       "  From Last  Same As Day  Odd  Even   1-10   11-20   21-30   31-40   41-50  \\\n",
       "0        40          NaN    2     4      3       1       0       1       1   \n",
       "1       NaN          NaN    2     4      1       3       0       2       0   \n",
       "2     10,25          NaN    3     3      1       0       3       0       2   \n",
       "\n",
       "   Division 1 Winners  Division 1 Prize  Division 2 Winners  Division 2 Prize  \\\n",
       "0                 1.0         3778784.0                 6.0           74237.0   \n",
       "1                 0.0               0.0                 4.0           87530.0   \n",
       "2                 2.0          774896.0                 1.0          326272.0   \n",
       "\n",
       "   Division 3 Winners  Division 3 Prize  Division 4 Winners  Division 4 Prize  \\\n",
       "0               217.0            1412.0               428.0             391.0   \n",
       "1               126.0            1911.0               380.0             346.0   \n",
       "2               157.0            1429.0               271.0             452.0   \n",
       "\n",
       "   Division 5 Winners  Division 5 Prize  Division 6 Winners  Division 6 Prize  \\\n",
       "0             10803.0              50.0             11247.0              25.0   \n",
       "1              7015.0              50.0              9381.0              25.0   \n",
       "2              6691.0              50.0              8285.0              25.0   \n",
       "\n",
       "   Division 7 Winners  Division 7 Prize  \n",
       "0            185785.0              10.0  \n",
       "1            136653.0              10.0  \n",
       "2            120531.0              10.0  "
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   "source": [
    "### Explore the first 3 rows of data.\n",
    "\n",
    "data.head(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
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       "      <td>2008-07-03</td>\n",
       "      <td>6</td>\n",
       "      <td>11</td>\n",
       "      <td>14</td>\n",
       "      <td>15</td>\n",
       "      <td>28</td>\n",
       "      <td>45</td>\n",
       "      <td>35</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      Draw        Date  Winning Number 1   2   3   4   5   6  \\\n",
       "1293  2343  2008-07-10                22  32  41  42  43  44   \n",
       "1294  2342  2008-07-07                 2  18  19  21  39  45   \n",
       "1295  2341  2008-07-03                 6  11  14  15  28  45   \n",
       "\n",
       "      Additional Number  From Last  Same As Day  Odd  Even   1-10   11-20  \\\n",
       "1293                  31       NaN          NaN    2     4      0       0   \n",
       "1294                  36        45          NaN    4     2      1       2   \n",
       "1295                  35       NaN          NaN    3     3      1       3   \n",
       "\n",
       "       21-30   31-40   41-50  Division 1 Winners  Division 1 Prize  \\\n",
       "1293       1       1       4                 NaN               NaN   \n",
       "1294       1       1       1                 NaN               NaN   \n",
       "1295       1       0       1                 NaN               NaN   \n",
       "\n",
       "      Division 2 Winners  Division 2 Prize  Division 3 Winners  \\\n",
       "1293                 NaN               NaN                 NaN   \n",
       "1294                 NaN               NaN                 NaN   \n",
       "1295                 NaN               NaN                 NaN   \n",
       "\n",
       "      Division 3 Prize  Division 4 Winners  Division 4 Prize  \\\n",
       "1293               NaN                 NaN               NaN   \n",
       "1294               NaN                 NaN               NaN   \n",
       "1295               NaN                 NaN               NaN   \n",
       "\n",
       "      Division 5 Winners  Division 5 Prize  Division 6 Winners  \\\n",
       "1293                 NaN               NaN                 NaN   \n",
       "1294                 NaN               NaN                 NaN   \n",
       "1295                 NaN               NaN                 NaN   \n",
       "\n",
       "      Division 6 Prize  Division 7 Winners  Division 7 Prize  \n",
       "1293               NaN                 NaN               NaN  \n",
       "1294               NaN                 NaN               NaN  \n",
       "1295               NaN                 NaN               NaN  "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Explore the last 3 rows of data.\n",
    "\n",
    "data.tail(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Function for Historical Data Check\n",
    "\n",
    "Below we're writing the function that will enable users to compare their ticket against the historical TOTO lottery data and determine whether they would have ever won by now.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0       {1, 4, 6, 40, 44, 15}\n",
       "1     {34, 40, 9, 13, 18, 20}\n",
       "2    {10, 46, 49, 23, 25, 28}\n",
       "3     {34, 9, 10, 42, 25, 27}\n",
       "4      {8, 9, 11, 47, 24, 26}\n",
       "dtype: object"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Function to extract all the winning six numbers from the historical data set as Python sets.\n",
    "\n",
    "def extract_numbers(row):\n",
    "    ### Get data from column index 2 to 7 (i.e. columns for 'Winning Number 1', '2', ... to '6').\n",
    "    row = row[2:8] \n",
    "    \n",
    "    ### Convert the values to 'set' datatype, i.e. a collection of unique items which is unordered and unindexed.\n",
    "    ### The major advantage of using a set, as opposed to a list, is that it has a highly optimized method \n",
    "    ###    for checking whether a specific element is contained in the set.\n",
    "    row = set(row.values)\n",
    "    return row\n",
    "\n",
    "### Apply the function to every row of the dataset (axis=1) to extract all the Winning Numbers.\n",
    "\n",
    "winning_numbers = data.apply(extract_numbers, axis=1) \n",
    "\n",
    "### Preview of the first 5 rows of the Winning Numbers.\n",
    "\n",
    "winning_numbers.head()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "### Function that takes in two inputs: \n",
    "### - a Python list containing the user numbers \n",
    "### - and a pandas Series containing sets with the Winning Numbers \n",
    "\n",
    "def check_historical_occurence(list_of_6_numbers, series_of_winning_numbers):\n",
    "    \n",
    "    ### Convert the user numbers list as a set.\n",
    "    input_numbers = set(list_of_6_numbers)\n",
    "    \n",
    "    ### Compare the set against the pandas Series that contains the sets with the Winning Numbers to find the number of matches.\n",
    "    ### A Series of Boolean values will be returned as a result of the comparison.\n",
    "    matches = series_of_winning_numbers == input_numbers\n",
    "    \n",
    "    ### Sum the matches to get the number of times the matches has occurred.\n",
    "    total_matches = matches.sum()\n",
    "    \n",
    "    ### Print information about the number of times the combination inputted by the user occurred in the past.    \n",
    "    template = \"This combination of numbers {input} has occurred {num1} time(s) as Winning Numbers in the past.\"\n",
    "    output = template.format(input=list_of_6_numbers, num1=int(total_matches))    \n",
    "    print(output)\n",
    "    \n",
    "    if total_matches == 0:\n",
    "        print(\"This doesn't mean it's more likely to occur now.\")    \n",
    "    \n",
    "    print('\\nIn the next drawing:')\n",
    "    \n",
    "    ### Call the one_ticket_probability() function again to print the probability of winning the Jackpot prize \n",
    "    ### in the next drawing with that combination.\n",
    "    one_ticket_probability(list_of_6_numbers) \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This combination of numbers [2, 41, 11, 12, 43, 14] has occurred 0 time(s) as Winning Numbers in the past.\n",
      "This doesn't mean it's more likely to occur now.\n",
      "\n",
      "In the next drawing:\n",
      "Your chances to win the Jackpot prize with the numbers [2, 41, 11, 12, 43, 14] are 0.0000072%.\n",
      "In other words, you have a 1 in 13,983,816 chances to win.\n"
     ]
    }
   ],
   "source": [
    "### Test the check_historical_occurence() function.\n",
    "\n",
    "check_historical_occurence([2, 41, 11, 12, 43, 14], winning_numbers)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This combination of numbers [6, 11, 14, 15, 28, 45] has occurred 1 time(s) as Winning Numbers in the past.\n",
      "\n",
      "In the next drawing:\n",
      "Your chances to win the Jackpot prize with the numbers [6, 11, 14, 15, 28, 45] are 0.0000072%.\n",
      "In other words, you have a 1 in 13,983,816 chances to win.\n"
     ]
    }
   ],
   "source": [
    "### Test the check_historical_occurence() function again with a known Winning Number.\n",
    "\n",
    "check_historical_occurence([6, 11, 14, 15, 28, 45], winning_numbers)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multi-Ticket Probability of Winning the Jackpot Prize\n",
    "\n",
    "Lottery addicts usually play more than one ticket on a single drawing, thinking that this might increase their chances of winning significantly. Our purpose is to help them better estimate their chances of winning — here, we're going to write a function that will calculate the chances of winning for any number of different tickets.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "### A function that prints the probability of winning the Jackpot prize depending on the number of different tickets played.\n",
    "\n",
    "def multi_ticket_probability(n_tickets):\n",
    "    \n",
    "    ### Start by calculating the total number of possible outcomes — this is total number of combinations \n",
    "    ###    for a six-number lottery ticket. \n",
    "    ### There are 49 possible numbers, and six numbers are sampled without replacement.\n",
    "    \n",
    "    n_combinations = combinations(49, 6)\n",
    "    \n",
    "    \n",
    "    ### Calculate the probability of winning the Jackpot prize based on the number of tickets.\n",
    "    \n",
    "    p_n_tickets = n_tickets / n_combinations\n",
    "    \n",
    "    \n",
    "    ### Convert the probability to percentage.\n",
    "    \n",
    "    percent = p_n_tickets * 100\n",
    "    \n",
    "    \n",
    "    ### Print the result in a friendly way.\n",
    "    \n",
    "    if n_tickets == 1:\n",
    "        template = \"\"\"Your chances to win the Jackpot prize with {input:,} ticket are {num1:.7f}%.\n",
    "In other words, you have a 1 in {num2:,} chances to win.\"\"\"\n",
    "    else:\n",
    "        template = \"\"\"Your chances to win the Jackpot prize with {input:,} different tickets are {num1:.7f}%.\n",
    "In other words, you have a 1 in {num2:,} chances to win.\"\"\"\n",
    "                \n",
    "    output = template.format(input=n_tickets, num1=percent, num2=round(n_combinations/n_tickets))\n",
    "    \n",
    "    print(output)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Your chances to win the Jackpot prize with 1 ticket are 0.0000072%.\n",
      "In other words, you have a 1 in 13,983,816 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n",
      "Your chances to win the Jackpot prize with 10 different tickets are 0.0000715%.\n",
      "In other words, you have a 1 in 1,398,382 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n",
      "Your chances to win the Jackpot prize with 100 different tickets are 0.0007151%.\n",
      "In other words, you have a 1 in 139,838 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n",
      "Your chances to win the Jackpot prize with 10,000 different tickets are 0.0715112%.\n",
      "In other words, you have a 1 in 1,398 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n",
      "Your chances to win the Jackpot prize with 1,000,000 different tickets are 7.1511238%.\n",
      "In other words, you have a 1 in 14 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n",
      "Your chances to win the Jackpot prize with 6,991,908 different tickets are 50.0000000%.\n",
      "In other words, you have a 1 in 2 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n",
      "Your chances to win the Jackpot prize with 13,983,816 different tickets are 100.0000000%.\n",
      "In other words, you have a 1 in 1 chances to win.\n",
      "-----------------------------------------------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "### Test the above function using the following test inputs.\n",
    "\n",
    "number_of_tickets = [1, 10, 100, 10000, 1000000, 6991908, 13983816]\n",
    "\n",
    "for n in number_of_tickets:\n",
    "    multi_ticket_probability(n)\n",
    "    print('-----------------------------------------------------------------------------------------')\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Less Winning Numbers - Probability of Winning Smaller Prizes\n",
    "\n",
    "For extra context, in [TOTO](https://online.singaporepools.com/en/lottery/toto-statistics-history) lottery there are smaller prizes if a player's ticket match three, four, or five of the six Winning Numbers drawn (or also match the Additional Number drawn). As a consequence, the users might be interested in knowing the probability of having three, four, or five Winning Numbers.\n",
    "\n",
    "- Group 2 Prize: Matching 5 Winning Numbers + Additional Number\n",
    "- Group 3 Prize: Matching 5 Winning Numbers\n",
    "- Group 4 Prize: Matching 4 Winning Numbers + Additional Number\n",
    "- Group 5 Prize: Matching 4 Winning Numbers \n",
    "- Group 6 Prize: Matching 3 Winning Numbers + Additional Number\n",
    "- Group 7 Prize: Matching 3 Winning Numbers \n",
    "\n",
    "Below are functions to allow users to calculate probabilities for three, four, or five Winning Numbers (for Group 3, 5, and 7 Prizes), and also with matching the Additional Number (for Group 2, 4, and 6 Prizes)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "### A function which takes in an integer n (value between 3 and 5) and prints information about the chances of winning \n",
    "### the Group 3, Group 5, or Group 7 prize, depending on the value of that integer.\n",
    "\n",
    "def probability_win_group357(n):\n",
    "    \n",
    "    ### Calculate the number of successful outcomes given the value of the input. \n",
    "        \n",
    "    ### Find the total number of n-number combinations from the set of 6 numbers (i.e. 6 choose n)\n",
    "    combinations_n = combinations(6, n)\n",
    "    \n",
    "    ### Find the remaining number of combinations (i.e. 42 choose 6-n).\n",
    "    ### Note: 42 is used instead of 43 because there is one number drawn for Additional Number.\n",
    "    combinations_remaining = combinations(49-6-1, 6-n)\n",
    " \n",
    "    ### Calculate the total number of successful outcomes.\n",
    "    ### (Imagine if n=3, we need to find combination from the ticket [6 choose 3] and the remaining combination [42 choose 3]).\n",
    "    ### (Imagine if n=4, we need to find combination from the ticket [6 choose 4] and the remaining combination [42 choose 2]).\n",
    "    ### (Imagine if n=5, we need to find combination from the ticket [6 choose 5] and the remaining combination [42 choose 1]).\n",
    "    total_successful_outcome = (combinations_n * combinations_remaining)\n",
    "    \n",
    "    ### Get the total possible outcomes from [49 choose 6] for Winning Numbers.\n",
    "    total_possible_outcome = combinations(49, 6)\n",
    "    \n",
    "    ### Calculate the probability\n",
    "    probability = total_successful_outcome / total_possible_outcome\n",
    "    \n",
    "    percent = probability * 100\n",
    "    \n",
    "    template = \"\"\"Your chances of having {input} Winning Numbers with this ticket are {num1:.7f}%.\n",
    "In other words, you have a 1 in {num2:,} chances to win.\"\"\"\n",
    "    \n",
    "    output = template.format(input=n, num1=percent, num2=int(round(total_possible_outcome / total_successful_outcome)))\n",
    "    \n",
    "    print(output)    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "### A function which takes in an integer n (value between 3 and 5) and prints information about the chances of winning \n",
    "### the Group 2, Group 4, or Group 6 prize, depending on the value of that integer.\n",
    "\n",
    "def probability_win_group246(n):\n",
    "    \n",
    "    ### Calculate the number of successful outcomes given the value of the input. \n",
    "    \n",
    "    ### Find the total number of n-number combinations from the set of 6 numbers (i.e. 6 choose n)\n",
    "    combinations_n = combinations(6, n)\n",
    "    \n",
    "    ### Find the remaining number of combinations (i.e. 42 choose 6-n-1).\n",
    "    ### Note: 42 is used instead of 43 because there is one number drawn for Additional Number.\n",
    "    combinations_remaining1 = combinations(49-6-1, 6-n-1)\n",
    "    \n",
    "    ### Combination [1 choose 1] is referring to the Additonal Number.\n",
    "    combinations_remaining2 = combinations(1, 1)\n",
    " \n",
    "    ### Calculate the total number of successful outcomes.\n",
    "    ### (Imagine if n=3, find combination from the ticket [6 choose 3] and remaining combination [42 choose 2] and [1 choose 1]).\n",
    "    ### (Imagine if n=4, find combination from the ticket [6 choose 4] and remaining combination [42 choose 1] and [1 choose 1]).\n",
    "    ### (Imagine if n=5, find combination from the ticket [6 choose 5] and remaining combination [42 choose 0] and [1 choose 1]).\n",
    "    total_successful_outcome = (combinations_n * combinations_remaining1 * combinations_remaining2)\n",
    "    \n",
    "    ### Get the total possible outcomes from [49 choose 6] for Winning Numbers and [43 choose 1] for the Additional Number.\n",
    "    total_possible_outcome = combinations(49, 6) + combinations(43, 1)\n",
    "        \n",
    "    ### Calculate the probability\n",
    "    probability = total_successful_outcome / total_possible_outcome\n",
    "    \n",
    "    percent = probability * 100\n",
    "    \n",
    "    template = \"\"\"Your chances of having {input} Winning Numbers + Additional Number with this ticket are {num1:.7f}%.\n",
    "In other words, you have a 1 in {num2:,} chances to win.\"\"\"\n",
    "    \n",
    "    output = template.format(input=n, num1=percent, num2=int(round(total_possible_outcome / total_successful_outcome)))\n",
    "    \n",
    "    print(output)    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Your chances of having 3 Winning Numbers with this ticket are 1.6418980%.\n",
      "In other words, you have a 1 in 61 chances to win.\n",
      "------------------------------------------------------------------------------------------\n",
      "Your chances of having 3 Winning Numbers + Additional Number with this ticket are 0.1231420%.\n",
      "In other words, you have a 1 in 812 chances to win.\n",
      "------------------------------------------------------------------------------------------\n",
      "Your chances of having 4 Winning Numbers with this ticket are 0.0923568%.\n",
      "In other words, you have a 1 in 1,083 chances to win.\n",
      "------------------------------------------------------------------------------------------\n",
      "Your chances of having 4 Winning Numbers + Additional Number with this ticket are 0.0045052%.\n",
      "In other words, you have a 1 in 22,197 chances to win.\n",
      "------------------------------------------------------------------------------------------\n",
      "Your chances of having 5 Winning Numbers with this ticket are 0.0018021%.\n",
      "In other words, you have a 1 in 55,491 chances to win.\n",
      "------------------------------------------------------------------------------------------\n",
      "Your chances of having 5 Winning Numbers + Additional Number with this ticket are 0.0000429%.\n",
      "In other words, you have a 1 in 2,330,643 chances to win.\n",
      "------------------------------------------------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "### Test the above functions using the following test inputs.\n",
    "\n",
    "test_inputs = [3, 4, 5]\n",
    "\n",
    "for n in test_inputs:\n",
    "    probability_win_group357(n)\n",
    "    print('------------------------------------------------------------------------------------------')\n",
    "    probability_win_group246(n)\n",
    "    print('------------------------------------------------------------------------------------------')\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Interested to view the frequencies of numbers drawn and the top 20 Jackpot prizes?**\n",
    "\n",
    "**[Click here](https://public.tableau.com/profile/peggy.chang#!/vizhome/TOTOLottery/S1) to check out these visualizations created with Tableau Public.**\n",
    "\n",
    "![Toto](Toto.png)"
   ]
  }
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