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{ "cells": [ { "cell_type": "markdown", "id": "4f43f640-f9bc-4919-8de8-811f4b2b7743", "metadata": {}, "source": [ "# Work assignment SW10 - PYTHON BASICS\n", "\n", "These are the self-study tasks of the semester week, which you will solve within one week in your JupyterHub environment. After completing your work, download a copy of the Jupyter notebook file locally to your laptop (Menu: File->Download).\n", "\n", "On ILIAS you will find the weekly scheduled assignment where you will upload your solved Jupyter notebook file. After your submission, you will receive a corresponding sample solution to the assignment. Your submission will not be corrected. Although the assignments are marked “mandatory”, they do not count towards your semester grades. Only the grades of the tests during the semester are relevant for this. \n", "\n", "We wish you every success!\n" ] }, { "cell_type": "markdown", "id": "6c156f95", "metadata": {}, "source": [ "---\n", "---\n", "## Exercise 1\n", "a) Simulate $1000$ tosses of an *unfair* coin, i.e. one in which $0$ and $1$ do not fall equally often. Specifically\n", "\n", "$$\n", "P(0)=0.3 \\qquad \\text{and} \\qquad P(1)=0.7\n", "$$\n", "\n", "shall apply. As parameter `p` of the method `numpy.choice()` you can give an array with the probabilities.\n", "\n", "### Solution:" ] }, { "cell_type": "code", "execution_count": 121, "id": "c9c5bb62", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1: [0.268 0.732]\n", "2: [0.299 0.701]\n", "3: [0.325 0.675]\n", "4: [0.274 0.726]\n", "5: [0.318 0.682]\n", "6: [0.306 0.694]\n", "7: [0.313 0.687]\n", "8: [0.29 0.71]\n", "9: [0.292 0.708]\n", "[1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1\n", " 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1\n", " 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1\n", " 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1\n", " 0 1 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0\n", " 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1\n", " 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0\n", " 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1\n", " 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0\n", " 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 0 1 0\n", " 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1\n", " 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 0 1 1\n", " 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1\n", " 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 1 0 1 1 1 1\n", " 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1\n", " 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1\n", " 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 0 1 1 0 0 1\n", " 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 0 1 0 0 1\n", " 1 1 1 0 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1\n", " 0 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1\n", " 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1\n", " 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1\n", " 1 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1\n", " 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 1 0\n", " 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0\n", " 1 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 0\n", " 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1\n", " 0]\n" ] } ], "source": [ "import numpy as np\n", "\n", "n_tosses = 1000\n", "coin_probabilities = [0.3,0.7]\n", "\n", "for elem in range(9):\n", " coin_toss = np.random.choice([0,1], size=n_tosses, p=coin_probabilities)\n", "\n", " rel_frequ_coin = np.bincount(coin_toss) / len(coin_toss)\n", "\n", " print(f\"{elem+1}: {rel_frequ_coin}\")\n", "\n", "print(coin_toss)" ] }, { "cell_type": "markdown", "id": "61039ff5", "metadata": {}, "source": [ "b) Plot a bar chart with the relative frequencies. Does the result meet your expectations?\n", "\n", "### Solution:" ] }, { "cell_type": "code", "execution_count": 132, "id": "92f7cc16", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "<Figure size 640x480 with 1 Axes>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "fig, ax = plt.subplots()\n", "ax.bar(['0','1'],rel_frequ_coin)\n", "plt.title('Real coin toss')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "d3f0d56f", "metadata": {}, "source": [ "c) Take the consecutive differences of the sequence and plot a bar chart of their relative frequencies again. Think qualitatively about why you get this result. We will learn the exact calculation later.\n", "\n", "### Solution:" ] }, { "cell_type": "code", "execution_count": 141, "id": "24719081", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "<Figure size 640x480 with 1 Axes>" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "diff_coin_toss = np.diff(coin_toss)\n", "rel_frequ_diff_coin = np.bincount(diff_coin_toss+1)/len(diff_coin_toss)\n", "\n", "fig, ax = plt.subplots()\n", "ax.bar(['-1','0','1'],rel_frequ_diff_coin)\n", "plt.title('Differences of the coin toss sequence')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "bb3fdd62-b44d-4d49-91e4-845c30ca8fe5", "metadata": {}, "source": [ "## Exercise 2\n", "A die has been rolled in such a way that every odd number appears twice as likely as every even number. All even numbers are equally likely. The die is rolled once.\n", "\n", "a) Describe the event space in terms of elementary events.\n", "\n", "### Solution:" ] }, { "cell_type": "markdown", "id": "7a455c8d-f4c9-45f2-a88e-f1d5b665c421", "metadata": {}, "source": [ "Dice faces: $\\{1,2,3,4,5,6\\}$\n", "\n", "$\\mathcal{F}_{\\text{odd}} = \\{1,3,5\\}$\n", "\n", "$\\mathcal{F}_{\\text{even}} = \\{2,4,6\\}$\n", "\n", "$\\mathcal{F}_{\\text{odd}} = \\frac{1}{2}\\cdot\\mathcal{F}_{\\text{even}}$\n", "\n", "$\\Omega = \\{1,1,2,3,3,4,5,5,6\\}$" ] }, { "cell_type": "markdown", "id": "f1c2f726-87a4-46ac-b2b1-fcbc241cdbff", "metadata": {}, "source": [ "b) Calculate the probability for each elementary event.\n", "\n", "### Solution:" ] }, { "cell_type": "markdown", "id": "afe4c3d8-9dc1-4b77-b488-67ed4504b2c4", "metadata": {}, "source": [ "$\\mathbb{P}(1,3,5) = \\frac{6}{9} = \\frac{2}{3}$\n", "\n", "$\\mathbb{P}(2,4,6) = \\frac{3}{9} = \\frac{1}{3}$" ] }, { "cell_type": "markdown", "id": "895370e8-5d10-42b8-ac98-ef98d0267a27", "metadata": {}, "source": [ "c) Calculate the probability that a number smaller than $4$ is rolled.\n", "\n", "### Solution:" ] }, { "cell_type": "markdown", "id": "e16c98e9-e84f-4e78-97c4-cdd7d7be7b0b", "metadata": {}, "source": [ "$\\mathcal{F}_{<4} = \\{(1,\\frac{2}{3}),(2,\\frac{1}{3}),(3,\\frac{2}{3})\\}$\n", "\n", "$\\mathbb{P}(\\mathcal{F}_{<4}) = \\mathcal{F}_{\\Omega} - \\overline{\\mathcal{F}}_{<4}$\n", "\n", "$\\mathbb{P}(\\mathcal{F}_{<4}) = \\frac{9}{9} - \\frac{4}{9} = \\frac{5}{9}$" ] }, { "cell_type": "markdown", "id": "ca916135-4b38-4bfb-b108-bd42c2cea658", "metadata": {}, "source": [ "d) Simulate that this die is rolled a $1000$ times to check the plausibility of your results from parts b) and c).\n", "\n", "### Solution:" ] }, { "cell_type": "code", "execution_count": 246, "id": "df997923-bfd9-4e7d-9f83-2f106b8bb43e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1: 20.4 %\n", "2: 12.7 %\n", "3: 19.6 %\n", "4: 12.0 %\n", "5: 22.4 %\n", "6: 12.9 %\n" ] } ], "source": [ "n_tosses = 1000\n", "dice_probabilities = [2/9,1/9,2/9,1/9,2/9,1/9]\n", "\n", "dice_toss = np.random.choice([1,2,3,4,5,6], size=n_tosses, p=dice_probabilities)\n", "\n", "rel_frequ_dice = np.bincount(dice_toss) / len(dice_toss)\n", "\n", "for x,elem in enumerate(rel_frequ_dice*100):\n", " if elem > 0: \n", " print(f\"{x}: {elem:.1f} %\")" ] }, { "cell_type": "markdown", "id": "22f560d7-860f-4a9b-a4cd-491691fbd3f5", "metadata": {}, "source": [ "## Exercise 3\n", "\n", "a) There are $56$ people at a birthday party. What is the probability that at least one other person will have a birthday that day?\n", "\n", "b) How many people must be at the party to have at least a $50\\%$ chance of another person having a birthday that day?\n", "\n", "c) Suppose there are $56$ people in a room. What is the probability that at least two people have a birthday on the same day?\n", "\n", "### Solution:" ] }, { "cell_type": "markdown", "id": "a9224fd2-fe08-4f17-80d7-79abc061f1dd", "metadata": {}, "source": [ "a) Combination with repetition:\n", "\n", "$\\mathbb{P}(\\text{birthday}) = 1 - \\mathbb{P}(\\text{no birthday})$\n", "\n", "$\\mathbb{P}(\\text{no birthday}) = \\dfrac{365-364-...-(365-56+1)}{365} = \\dfrac{365!}{(365-56)!\\cdot 365^{56}}$\n", "\n", "$\\mathbb{P}(\\text{birthday}) = 1 - \\dfrac{365!}{(365-56)!\\cdot 365^{56}}$" ] }, { "cell_type": "code", "execution_count": 263, "id": "a9eacd67", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P(birthday) = 98.8%\n" ] } ], "source": [ "import math\n", "\n", "no_birthday = math.factorial(365)/(math.factorial(365-56)*365**56)\n", "\n", "birthday = 1 - no_birthday\n", "\n", "print(f\"P(birthday) = {round(birthday*100,1)}%\")" ] }, { "cell_type": "markdown", "id": "67255485", "metadata": {}, "source": [ "b)\n", "\n", "c)" ] }, { "cell_type": "markdown", "id": "f9c5548e-6d5d-4a28-b107-751c2a221474", "metadata": {}, "source": [ "## Exercise 4\n", "Poker is played with the Anglo-American hand, which comprises $13$ cards of different values from the $4$ suits of diamonds, hearts, spades and clubs: $2$ to $10$, jack, queen, king and ace. Each player is dealt $5$ cards at random. We assume that each card has the same probability. Calculate the probability for the following hands:\n", "\n", "a) **Royal Flush**: $10$, Jack, Queen, King, Ace of the same suit\n", "\n", "b) **Four of a kind**: Four cards of the same value\n", "\n", "c) **Full House**: A three of a kind (three cards of the same value) and a pair (two cards of the same value).\n", "\n", "### Solution:" ] }, { "cell_type": "markdown", "id": "51c41ba6-76af-47e5-a921-4c693851cf3a", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": null, "id": "e389fee0-14d9-4e07-8a72-ca261c3f3ac2", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.11.9" } }, "nbformat": 4, "nbformat_minor": 5 }
LUNAR