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April 7, 2019 17:58
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Aufgabe 26 UE Wahrscheinlichkeitstheorie
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| "# Aufgabe 26\n", | |
| "\n", | |
| "Eine Münze wird viermal hintereinander geworfen. Betrachten wir den Grundraum $\\Omega=\\{K,Z\\}^4$, den Wahrscheinlichkeitsraum $(\\Omega, \\mathcal{P}(\\Omega), \\mathbb{P}_u)$ und die\n", | |
| "Ereignisse $A_j =$ \\{Kopf tritt j-mal ein\\}, $j = 0, 1, 2, 3, 4$.\n", | |
| "\n", | |
| "(a) Bestimmen Sie die Wahrscheinlichkeiten $\\mathbb{P}(A_j)$, $j = 0, 1, 2, 3, 4$, und zeichnen Sie ein Stabdiagramm der Verteilung von der Zufallsvariable $X(A_j) = j$.\n", | |
| "\n", | |
| "(b) Wie ändert sich das Stabdiagramm, wenn die Münze nicht homogen (regelmäßig, echt, fair) ist? Nehmen Sie an, dass die Wahrscheinlichkeit für Kopf $p = 0.4$ ist.\n", | |
| "***" | |
| ] | |
| }, | |
| { | |
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| "execution_count": 1, | |
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| "source": [ | |
| "import random\n", | |
| "import plotly.offline as pl # Für die Säulendiagramme\n", | |
| "import plotly.graph_objs as go # Für die Säulendiagramme" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "def simulation(N=1000000, fair=True):\n", | |
| " muenze = ('K', 'Z') # Eine Muenze besteht aus 'K'opf und 'Z'ahl\n", | |
| " anzahl_kopf = [0]*5 # Wie oft kommt 0-5 mal Kopf vor\n", | |
| " if fair:\n", | |
| " weights = None\n", | |
| " else:\n", | |
| " weights = (0.4, 0.6) # Kopf hat Wahrscheinlichkeit p=0.4, Zahl hat p=0.6\n", | |
| " for _ in range(N):\n", | |
| " wurf = random.choices(muenze, weights, k=4)\n", | |
| " anzahl_kopf[wurf.count('K')] += 1\n", | |
| " return [x/N for x in anzahl_kopf]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
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| "text": [ | |
| "Faire Münze: [0.062293, 0.249993, 0.374244, 0.250522, 0.062948]\n", | |
| "Kopf p=0.4: [0.129555, 0.34569, 0.345465, 0.153553, 0.025737]\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "N = 1000000\n", | |
| "anzahl_fair = simulation(N)\n", | |
| "print(\"Faire Münze:\", anzahl_fair)\n", | |
| "anzahl_unfair = simulation(N, fair=False)\n", | |
| "print(\"Kopf p=0.4:\", anzahl_unfair)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Säulendiagramme für die relativen Häufigkeiten der beiden Münzen (fair/unfair)." | |
| ] | |
| }, | |
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| "execution_count": 4, | |
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| 0.062293, | |
| 0.249993, | |
| 0.374244, | |
| 0.250522, | |
| 0.062948 | |
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| 4 | |
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| 0.129555, | |
| 0.34569, | |
| 0.345465, | |
| 0.153553, | |
| 0.025737 | |
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| "text": "Anzahl Kopf" | |
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| "source": [ | |
| "pl.init_notebook_mode(connected=True)\n", | |
| "layout = {\n", | |
| " 'xaxis': {'title': 'Anzahl Kopf'},\n", | |
| " 'yaxis': {'title': 'Relative Häufigkeit'},\n", | |
| " 'barmode': 'group',\n", | |
| " 'title': 'Relative Häufigkeiten für die Anzahl an \"Kopf\" beim 4 fachen Münzwurf bei einer Simulation von %d Runden ' % N\n", | |
| "}\n", | |
| "data1 = go.Bar(\n", | |
| " x = list(range(5)),\n", | |
| " y = anzahl_fair,\n", | |
| " name = 'Fair'\n", | |
| " )\n", | |
| "data2 = go.Bar(\n", | |
| " x = list(range(5)),\n", | |
| " y = anzahl_unfair,\n", | |
| " name = 'Unfair'\n", | |
| " )\n", | |
| "data=[data1, data2]\n", | |
| "fig = go.FigureWidget(data, layout)\n", | |
| "pl.iplot(fig)" | |
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