S03b_Exercises1_solution.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Execute before using this notebook if using google colab\n",
    "\n",
    "kernel = str(get_ipython())\n",
    "\n",
    "if 'google.colab' in kernel:    \n",
    "    !wget https://raw.githubusercontent.com/fredzett/rmqa/master/utils.py -P local_modules -nc \n",
    "    !npx degit fredzett/rmqa/data data\n",
    "    import sys\n",
    "    sys.path.append('local_modules')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from utils import plot_line, plot_bar, plot_hist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the probability when rolling three dices that the sum of eyes of those three eyes is $\\geq 15$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## Solution 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 4, 1],\n",
       "       [4, 2, 3],\n",
       "       [2, 6, 1],\n",
       "       ...,\n",
       "       [5, 4, 4],\n",
       "       [4, 4, 1],\n",
       "       [6, 5, 1]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.randint(1,7,(n, 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.09191"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 100000\n",
    "dices = np.random.randint(1,7,size=(n,3))\n",
    "sum(np.sum(dices,axis=1) >= 15)/n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Philip Founder has invested $200.000$ in his new business. Based on research he expects the following number of customers per period:\n",
    "\n",
    "- customers: 10, 40, 60, 100 \n",
    "- probability of arrival: 10%, 50%, 30%, 10%\n",
    "\n",
    "Example: with a probability of 50% he expects 40 customers a day \n",
    "\n",
    "On average his after cost cashflow is $120$ EUR per customer. \n",
    "\n",
    "Assume that the business will last for 50 periods. \n",
    "\n",
    "What is the probability of recovering his investment after:\n",
    "\n",
    "- 26 periods\n",
    "- 30 periods\n",
    "- 40 periods\n",
    "\n",
    "(Tip: use `utils.plot_bar` or `utils.plot_hist`)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_earnings():\n",
    "    'Simulates business for 50 periods and returns cumulative earnings per period'\n",
    "    choices = [10,40,60,100]\n",
    "    probs = [0.1,0.5,0.3,0.1]\n",
    "    price_customer = 120\n",
    "\n",
    "    periods = 50\n",
    "    earnings = np.empty(periods)\n",
    "    for i in range(periods):\n",
    "        customer = np.random.choice(choices, p=probs)\n",
    "        earnings[i] = customer * price_customer\n",
    "    return np.cumsum(earnings)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "35"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.argwhere(simulate_earnings() >= 200000)[0][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 10000\n",
    "investment = 200_000\n",
    "break_even = []\n",
    "for i in range(n):\n",
    "    break_even.append(np.argwhere(simulate_earnings() >= investment)[0][0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "33.5753"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(break_even)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 300,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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   "source": [
    "days, amounts = np.unique(break_even, return_counts=True)\n",
    "probs = amounts / n\n",
    "plot_bar(days,probs);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_hist(break_even, show_prob=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "34.01360544217687"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "customers = 0.1*10+40*0.5+60*0.3+100*0.1\n",
    "200000 / (customers * 120)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You play a game with your friend. The game works as follows:\n",
    "\n",
    "- You filp a coin. Head = you win, Tail = your friend wins  \n",
    "    *(assume that it is a fair coin with $50\\%$ probability of head or tail)*\n",
    "\n",
    "- The person that wins 4 flips wins the overall game\n",
    "\n",
    "Questions:\n",
    "\n",
    "- what is the probability that the game lasts 4 games\n",
    "\n",
    "- what is the probability that the game lasts 5 games\n",
    "\n",
    "- what is the probability that the game lasts 7 games"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1217"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# game lasts 4 games\n",
    "n = 10000\n",
    "games = np.random.randint(0,2, size=(n,4))\n",
    "sums = np.sum(games, axis=1)\n",
    "(sum(sums == 4) + sum(sums==0)) / n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [],
   "source": [
    "# game lasts 4, 5, or 7 games\n",
    "\n",
    "def flip():\n",
    "    return np.random.randint(0,2)\n",
    "\n",
    "def game():\n",
    "    game = {\"A\":0, \"B\":0, \"stop\":False}\n",
    "\n",
    "    while game[\"stop\"] == False:\n",
    "        \n",
    "        if flip() == 0:\n",
    "            game[\"A\"] += 1\n",
    "        else:\n",
    "            game[\"B\"] +=1\n",
    "\n",
    "        if (game[\"A\"] == 4) | (game[\"B\"]==4):\n",
    "            game[\"stop\"] = True\n",
    "    \n",
    "    return game\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100000\n",
    "length = {4:0,5:0,6:0, 7:0}\n",
    "for i in range(n):\n",
    "    result = game()\n",
    "    games = result[\"A\"] + result[\"B\"]\n",
    "    length[games] += 1\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.12585, 0.24959, 0.31053)"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "length.get(4) / n, length.get(5) / n, length.get(7) / n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
 ],
 "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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  "toc-autonumbering": false
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