Counting Principles.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MSDS 400 Counting Principles\n",
    "\n",
    "The following is an example creates three functions used in counting principles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def factorial(n):\n",
    "    if n == 0:\n",
    "        return 1\n",
    "    else:\n",
    "        recurse = factorial(n - 1)\n",
    "        result = n * recurse\n",
    "        return result\n",
    "\n",
    "\n",
    "def perm(n, k):\n",
    "    if n == 0:\n",
    "        return 1\n",
    "    if k > n:\n",
    "        return -1\n",
    "    else:\n",
    "        return (factorial(n)) / factorial(n - k)\n",
    "\n",
    "\n",
    "def comb(n, k):\n",
    "    result = perm(n, k)\n",
    "    result = result / factorial(k)\n",
    "    result = int(result)\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Blewo you are prompted to enter an integer when you run the cell.\n",
    "\n",
    "Enter a positive integer to obtain the factorial value.<br>\n",
    "Enter a negative integer to stop.<br>\n",
    "In either case, hit enter or the code will not work.\n",
    "\n",
    "The statement `input()` accepts keyboard input and produces a string which must be converted to a numeric variable for further calculations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "What number should we get the factorial for?30\n",
      "Factorial of 30 is equal to 265252859812191058636308480000000\n"
     ]
    }
   ],
   "source": [
    "inp1 = input(\"What number should we get the factorial for?\")\n",
    "int_inp1 = int(inp1)\n",
    "\n",
    "if int_inp1 > 0:\n",
    "    fact_of_inp = factorial(int_inp1)\n",
    "    print('Factorial of {} is equal to {}'.format(int_inp1, fact_of_inp))\n",
    "else:\n",
    "    print('No calculation.')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Permutation of 10 elements taken 5 at a time =30240.0\n",
      "Combination of 10 elements taken 5 at a time = 252\n"
     ]
    }
   ],
   "source": [
    "# Example calculations follow.\n",
    "permutation = perm(10, 5)\n",
    "print('Permutation of 10 elements taken 5 at a time ={}'.format(permutation))\n",
    "\n",
    "combination = comb(10, 5)\n",
    "print('Combination of 10 elements taken 5 at a time = {}'.format(combination))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the code which follows, note the type conversions from string to integer, integer to floating point and back.\\\n",
    "\n",
    "In each instance, hit enter after submitting the requested number. Enter a positive integer for the number of repeated trials."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of Trials:5\n",
      "Number of Successes:8\n",
      "Probability of Success:2\n",
      "Binomial probability with n = 5, k= 8, p= 2.0 is -0.0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "inp = input(\"Number of Trials:\")\n",
    "int_inp = int(inp)  # This converts n from a string to an integer.\n",
    "\n",
    "success = input(\"Number of Successes:\")\n",
    "int_success_int = int(success)\n",
    "\n",
    "prob_of_s = input(\"Probability of Success:\")\n",
    "int_prob_of_s = float(prob_of_s)  # This converts p from a string to a floating point number.\n",
    "\n",
    "prob = (comb(int_inp, int_success_int)) *\\\n",
    "       (int_prob_of_s ** int_success_int) * \\\n",
    "       ((1.0 - int_prob_of_s) ** (int_inp - int_success_int))\n",
    "print('Binomial probability with n = {}, k= {}, p= {} is {}\\n'.format(int_inp, int_success_int, int_prob_of_s, prob))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following example shows how to calculate a binomial distribution\n",
    "and print the result.  First a binomial function will be defined.  The\n",
    "values previously entered for the number of repeated trials, the\n",
    "number of successes and the probability of success will be used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Binomial distribution with 5 trials and p=2.0 follows\n",
      "# of successes = 0 probability = -1.00\n",
      "# of successes = 1 probability = 10.00\n",
      "# of successes = 2 probability = -40.00\n",
      "# of successes = 3 probability = 80.00\n",
      "# of successes = 4 probability = -80.00\n",
      "# of successes = 5 probability = 32.00\n"
     ]
    }
   ],
   "source": [
    "def binomial(n, k, p):\n",
    "    return (comb(n, k)) * (p ** k) * ((1.0 - p) ** (n - k))\n",
    "\n",
    "\n",
    "print('Binomial distribution with {} trials and p={} follows'.format(int_inp, int_prob_of_s))\n",
    "distribution = []\n",
    "for i in range(0, int_inp + 1):\n",
    "    prob = binomial(int_inp, i, int_prob_of_s)\n",
    "    print('# of successes = {} probability = {:2.2f}'.format(i, prob))\n",
    "    distribution = distribution + [prob]\n",
    "\n",
    "x = range(0, int_inp + 1, 1)  # x and y are both lists with the same number of elements.\n",
    "y = distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot is produced in layers.  bar() places the red bars of width 0.4\n",
    "centered on x values. The heigth is determined by y.  The plot()\n",
    "statement places the line plot on the chart.  With only one show()\n",
    "statement, the figure() statement is unnecessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.figure()\n",
    "plt.bar(x, y, width=0.4, align='center', color='r')\n",
    "plt.plot(x, y)\n",
    "plt.xlabel('Total Number of Trials')\n",
    "plt.ylabel('Probability of Success')\n",
    "plt.title('Binomial Probabilities ')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the probability of randomly drawing five cards from a deck and getting exactly two Aces.\n",
    "The solution is similar to the previous example, except now we are choosing 2 Aces out of 4 and 3 non-Aces out of 48; the denominator remains the same:\n",
    "P(two Aces) = ((4C2 )*(48C3 ))/(52C5) = 103776/2598960 = ~ 0.0399"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def comb(n, k):\n",
    "    result = perm(n, k)\n",
    "    result = result / factorial(k)\n",
    "    result = int(result)\n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "P2A = (comb(4,2)*comb(48,3))/(comb(52,5)) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.03992981808107859"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The probability of randomly drawing five cards from a deck & getting exactly two Aces = ~ 0.0399\n"
     ]
    }
   ],
   "source": [
    "print('The probability of randomly drawing five cards from a deck & getting exactly two Aces = ~ {:.3}'.format(P2A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conditional Probability Formula\n",
    "If Events A and B are not independent, then P(A and B) = P(A) · P(B | A) "
   ]
  }
 ],
 "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",
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 },
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