Beispiel 21 UE Wahrscheinlichkeitstheorie

Beispiel 21.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulation von Lottospielen"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "import collections\n",
    "import pprint # Schöne Ausgabe von dictionaries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "****************************************************************************************************\n",
      "Kontrolle: 1000000\n",
      "defaultdict(<class 'int'>,\n",
      "            {'0': 338857,\n",
      "             '0Z': 61707,\n",
      "             '1': 368900,\n",
      "             '1Z': 54402,\n",
      "             '2': 136649,\n",
      "             '2Z': 15659,\n",
      "             '3': 20681,\n",
      "             '3Z': 1738,\n",
      "             '4': 1291,\n",
      "             '4Z': 72,\n",
      "             '5': 42,\n",
      "             '5Z': 1,\n",
      "             '6': 1})\n"
     ]
    }
   ],
   "source": [
    "def simulation(gewaehlte_zahlen, N=1000000):\n",
    "    '''Simuliert N (default=10^6) Lottoziehungen.\n",
    "    Gibt die Anzahl der jeweils richtig getippten Zahlen in einem dictionary zurück.'''\n",
    "    moegliche_zahlen = list(range(1, 46)) # Es sind die Zahlen 1-45 erlaubt\n",
    "    ergebnis = collections.defaultdict(int)\n",
    "    gewaehlte_zahlen = set(gewaehlte_zahlen)\n",
    "    for i in range(N):\n",
    "        if i%(N//100)==0: # Fortschrittsbalken; Gib 100 '*' aus um den Fortschritt der Simulation zu sehen\n",
    "            print(\"*\", end=\"\", flush=True)\n",
    "        ziehung = random.sample(moegliche_zahlen, 7) # Ziehe 6 Zahlen und eine Zusatzzahl\n",
    "        zusatzzahl = ziehung[6] # Zuletzt gewählte Zahl ist Zusatzzahl\n",
    "        ziehung = set(ziehung[:6]) # Ziehung besteht aus den ersten 6 Zahlen. \n",
    "        anzahl_richtiger_zahlen = len(ziehung & gewaehlte_zahlen)               \n",
    "        if zusatzzahl in gewaehlte_zahlen:\n",
    "            ergebnis['%dZ' % anzahl_richtiger_zahlen] += 1\n",
    "        else:\n",
    "            ergebnis[str(anzahl_richtiger_zahlen)] += 1\n",
    "    print()\n",
    "    return ergebnis\n",
    "\n",
    "ergebnis = simulation({1, 2, 3, 4, 5, 6}) # Da jede mögliche Folge an Zahlen gleichwahrscheinlich, wähle ich einfach die Zahlen 1 bis 6\n",
    "                                          # Simuliere 1 Million Lottospiele\n",
    "print(\"Kontrolle:\",sum(ergebnis.values()))\n",
    "pprint.pprint(ergebnis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ich habe die Simulation mit $10^{10}$ Lottorunden durchgeführt (das hat auf meinem PC einen Nachmittag benötigt) und das folgende Ergebnis erhalten:\n",
    "***\n",
    "\n",
    "    defaultdict(<class 'int'>,\n",
    "            {'0': 3389424243,\n",
    "             '0Z': 616274885,\n",
    "             '1': 3697478438,\n",
    "             '1Z': 543729648,\n",
    "             '2': 1359379079,\n",
    "             '2Z': 155365327,\n",
    "             '3': 207147579,\n",
    "             '3Z': 17264021,\n",
    "             '4': 12948012,\n",
    "             '4Z': 699918,\n",
    "             '5': 280333,\n",
    "             '5Z': 7351,\n",
    "             '6': 1166})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir wollen davon die relativen Häufigkeiten berechnen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "h( X = 0 ) = 0.3389424243\n",
      "h( X = 0Z ) = 0.0616274885\n",
      "h( X = 1 ) = 0.3697478438\n",
      "h( X = 1Z ) = 0.0543729648\n",
      "h( X = 2 ) = 0.1359379079\n",
      "h( X = 2Z ) = 0.0155365327\n",
      "h( X = 3 ) = 0.0207147579\n",
      "h( X = 3Z ) = 0.0017264021\n",
      "h( X = 4 ) = 0.0012948012\n",
      "h( X = 4Z ) = 6.99918e-05\n",
      "h( X = 5 ) = 2.80333e-05\n",
      "h( X = 5Z ) = 7.351e-07\n",
      "h( X = 6 ) = 1.166e-07\n"
     ]
    }
   ],
   "source": [
    "results = {'0': 3389424243,\n",
    "           '0Z': 616274885,\n",
    "           '1': 3697478438,\n",
    "           '1Z': 543729648,\n",
    "           '2': 1359379079,\n",
    "           '2Z': 155365327,\n",
    "           '3': 207147579,\n",
    "           '3Z': 17264021,\n",
    "           '4': 12948012,\n",
    "           '4Z': 699918,\n",
    "           '5': 280333,\n",
    "           '5Z': 7351,\n",
    "           '6': 1166}\n",
    "for k,v in results.items():\n",
    "    print('h( X =',k,') =',v/1e10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Theoretische Wahrscheinlichkeiten für die Ausgänge der Zufallsvariable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P( X = 0 )  = 0.338939\n",
      "P( X = 0Z ) = 0.0616253\n",
      "P( X = 1 )  = 0.369752\n",
      "P( X = 1Z ) = 0.0543753\n",
      "P( X = 2 )  = 0.135938\n",
      "P( X = 2Z ) = 0.0155358\n",
      "P( X = 3 )  = 0.0207144\n",
      "P( X = 3Z ) = 0.0017262\n",
      "P( X = 4 )  = 0.00129465\n",
      "P( X = 4Z ) = 6.99811e-05\n",
      "P( X = 5 )  = 2.79924e-05\n",
      "P( X = 5Z ) = 7.36643e-07\n",
      "P( X = 6 )  = 1.22774e-07\n"
     ]
    }
   ],
   "source": [
    "import operator as op\n",
    "from functools import reduce\n",
    "\n",
    "def ncr(n, r): # Binomialkoeffizient n über r\n",
    "    '''https://stackoverflow.com/a/4941932'''\n",
    "    r = min(r, n-r)\n",
    "    numer = reduce(op.mul, range(n, n-r, -1), 1)\n",
    "    denom = reduce(op.mul, range(1, r+1), 1)\n",
    "    return numer // denom\n",
    "\n",
    "for richtige in range(7): # 0 - 5Z\n",
    "    print('P( X = %d )  = %g' %(richtige, ncr(6,richtige)*ncr(39,6-richtige)/ncr(45,6)*(39-6+richtige)/39))\n",
    "    if richtige != 6:\n",
    "        print('P( X = %dZ ) = %g' %(richtige, ncr(6,richtige)*ncr(39,6-richtige)/ncr(45,6)*(6-richtige)/39))"
   ]
  }
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