{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# 集合と確率を操作する\n",
    "## 集合とは何か\n",
    "### 集合の構成"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: sympy in /usr/local/lib/python3.7/site-packages (1.3)\r\nRequirement already satisfied: mpmath>=0.19 in /usr/local/lib/python3.7/site-packages (from sympy) (1.0.0)\r\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "!{sys.executable} -m pip install sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{2, 4, 6}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(2, 4, 6)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1/5, 1, 1.5}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "from fractions import Fraction\n",
    "s = FiniteSet(1, 1.5, Fraction(1, 5))\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 1.5, 3)\n",
    "print(len(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ある数が集合にあるかどうかチェックする"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 1.5, 3)\n",
    "print(4 in s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 空集合を作る"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EmptySet()\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet()\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### リストやタプルから集合を作る"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2, 3}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "members = [1, 2, 3]\n",
    "s = FiniteSet(*members)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 集合要素の重複と順序"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2, 3}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "members = [1, 2, 3, 2]\n",
    "s = FiniteSet(*members)\n",
    "print(s)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n2\n3\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3)\n",
    "for member in s:\n",
    "    print(member)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(3, 4, 5)\n",
    "t = FiniteSet(5 ,4, 3)\n",
    "print(s == t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 部分集合、上位集合、べき集合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\nTrue\nTrue\nTrue\nFalse\nTrue\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "\n",
    "s = FiniteSet(1)\n",
    "t = FiniteSet(1, 2)\n",
    "print(s.is_subset(t))\n",
    "print(s.is_subset(s))\n",
    "print(s.is_subset(s))\n",
    "print(s.is_subset(t))\n",
    "\n",
    "print(s.is_superset(t))\n",
    "print(s.is_superset(s))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{EmptySet(), {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}\n8\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3)\n",
    "ps = s.powerset()\n",
    "print(ps)\n",
    "print(len(ps))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\nFalse\nTrue\nFalse\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3)\n",
    "t = FiniteSet(1, 2, 3)\n",
    "print(s.is_proper_subset(t))\n",
    "print(s.is_proper_subset(s))\n",
    "\n",
    "t = FiniteSet(1, 2, 3, 4)\n",
    "print(s.is_proper_subset(t))\n",
    "print(s.is_proper_subset(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 集合演算"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 和と積"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2, 3, 4, 6}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3)\n",
    "t = FiniteSet(2, 4, 6)\n",
    "print(s.union(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "${1, 2}\\cup{2, 3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{2}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2)\n",
    "t = FiniteSet(2, 3)\n",
    "print(s.intersect(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "${1, 2, 3}\\cup{2, 4, 6}\\cup{3, 5, 7}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2, 3, 4, 5, 6, 7}\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3)\n",
    "t = FiniteSet(2, 4, 6)\n",
    "u = FiniteSet(3, 5, 7)\n",
    "print(s.union(t).union(u))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "${1, 2, 3}\\cap{2, 4, 6}\\cap{3, 5, 7}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EmptySet()\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3)\n",
    "t = FiniteSet(2, 4, 6)\n",
    "u = FiniteSet(3, 5, 7)\n",
    "print(s.intersect(t).intersect(u))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 直積"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2} x {3, 4}\n(1, 3)\n(1, 4)\n(2, 3)\n(2, 4)\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2)\n",
    "t = FiniteSet(3, 4)\n",
    "p = s*t\n",
    "print(p)\n",
    "for elem in p:\n",
    "    print(elem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2} x {1, 2} x {1, 2}\n(1, 1, 1)\n(1, 1, 2)\n(1, 2, 1)\n(1, 2, 2)\n(2, 1, 1)\n(2, 1, 2)\n(2, 2, 1)\n(2, 2, 2)\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2)\n",
    "p = s**3\n",
    "print(p)\n",
    "for elem in p:\n",
    "    print(elem)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 変数を複数集合に公式を適用する"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$T = 2\\pi \\sqrt{\\frac{L}{g}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Length: 15.0 cm Time Period: 0.777 s\nLength: 18.0 cm Time Period: 0.852 s\nLength: 21.0 cm Time Period: 0.920 s\nLength: 22.5 cm Time Period: 0.952 s\nLength: 25.0 cm Time Period: 1.004 s\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet, pi\n",
    "\n",
    "\n",
    "def time_period(length):\n",
    "    g = 9.8\n",
    "    T = 2 * pi * (length / g) ** 0.5\n",
    "    return T\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    L = FiniteSet(15, 18, 21, 22.5, 25)\n",
    "    for l in L:\n",
    "        t = time_period(l / 100)\n",
    "        print('Length: {0} cm Time Period: {1:.3f} s'.format(float(l), float(t)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 異なる重力、異なる結果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Length(cm)   Gravity(m/s^2) Time Period(s) \n     15.0           9.78           0.201     \n     15.0            9.8           0.201     \n     15.0           9.83           0.201     \n     18.0           9.78           0.201     \n     18.0            9.8           0.201     \n     18.0           9.83           0.201     \n     21.0           9.78           0.201     \n     21.0            9.8           0.201     \n     21.0           9.83           0.201     \n     22.5           9.78           0.201     \n     22.5            9.8           0.201     \n     22.5           9.83           0.201     \n     25.0           9.78           0.201     \n     25.0            9.8           0.201     \n     25.0           9.83           0.201     \n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet, pi\n",
    "\n",
    "\n",
    "def time_period(length, g):\n",
    "    g = 9.8\n",
    "    T = 2*pi*(length/g)**0.5\n",
    "    return T\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    L = FiniteSet(15, 18, 21, 22.5, 25)\n",
    "    g_values = FiniteSet(9.8, 9.78, 9.83)\n",
    "    print('{0:^15}{1:^15}{2:^15}'.format('Length(cm)','Gravity(m/s^2)', 'Time Period(s)'))\n",
    "    for elem in L*g_values:\n",
    "        l = elem[0]\n",
    "        g = elem[1]\n",
    "        t = time_period(1/100, g)\n",
    "\n",
    "        print('{0:^15}{1:^15}{2:^15.3f}'.format(float(l),float(g),float(t)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 確率"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P(E)=\\frac{n(E)}{n(S)}$ $n(E)=8$かつ$n(S)=20$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample space: {1, 2, 3, ..., 18, 19, 20}\nEvent: {2, 3, 5, 7, 11, 13, 17, 19}\nProbability of rolling a prime:  0.40000\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "\n",
    "\n",
    "def probability(space, event):\n",
    "    return len(event)/len(space)\n",
    "\n",
    "\n",
    "def check_prime(number):\n",
    "    if number != 1:\n",
    "        for factor in range(2, number):\n",
    "            if number % factor == 0:\n",
    "                return False\n",
    "    else:\n",
    "        return False\n",
    "    return True\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    space = FiniteSet(*range(1,21))\n",
    "    primes = []\n",
    "    for num in space:\n",
    "        if check_prime(num):\n",
    "            primes.append(num)\n",
    "    event = FiniteSet(*primes)\n",
    "    p = probability(space, event)\n",
    "    \n",
    "    print('Sample space: {0}'.format(space))\n",
    "    print('Event: {0}'.format(event))\n",
    "    print('Probability of rolling a prime: {0: .5f}'.format(p))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 事象Aまたは事象Bの確率"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$E = {2, 3, 5} \\cup {1, 3, 5 } = { 1, 2, 3, 5}$\n",
    "\n",
    "$P(E) = \\frac{n(E)}{n(S)}=\\frac{4}{6}=\\frac{2}{3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6666666666666666\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3, 4, 5, 6)\n",
    "a = FiniteSet(2, 3, 5)\n",
    "b = FiniteSet(1, 3, 5)\n",
    "e = a.union(b)\n",
    "print(len(e)/len(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 事象Aおよび事象Bの確率"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$E = A \\cap B = {2, 3, 5} \\cup {1, 3, 5} = {3, 5}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.3333333333333333\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "s = FiniteSet(1, 2, 3, 4, 5, 6)\n",
    "a = FiniteSet(2, 3, 5)\n",
    "b = FiniteSet(1, 3, 5)\n",
    "e = a.intersect(b)\n",
    "print(len(e)/len(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 乱数生成\n",
    "#### サイコロ投げをシュミレーションする"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n4\n3\n3\n6\n4\n5\n3\n4\n5\n"
     ]
    }
   ],
   "source": [
    "import random\n",
    "for i in range(0, 10):\n",
    "    print(random.randint(1, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### その目を出せますか"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rolled: 6\nRolled: 2\nRolled: 4\nRolled: 5\nRolled: 3\nScore of 20 reached in 5 rolls\n"
     ]
    }
   ],
   "source": [
    "'''\n",
    "総和が20になるまでサイコロをふる\n",
    "'''\n",
    "import matplotlib.pyplot as plt\n",
    "import random\n",
    "\n",
    "target_score = 20\n",
    "\n",
    "\n",
    "def roll():\n",
    "    return random.randint(1, 6)\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    score = 0\n",
    "    num_rolls = 0\n",
    "    while score < target_score:\n",
    "        die_roll = roll()\n",
    "        num_rolls += 1\n",
    "        print('Rolled: {0}'.format(die_roll))\n",
    "        score += die_roll\n",
    "        \n",
    "print('Score of {0} reached in {1} rolls'.format(score, num_rolls))        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 目標点数は可能か"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probability: 0.00000\nProbability: 0.03241\n"
     ]
    }
   ],
   "source": [
    "from sympy import FiniteSet\n",
    "import random\n",
    "\n",
    "\n",
    "def find_prob(target_score, max_rolls):\n",
    "    die_sides = FiniteSet(1, 2, 3, 4, 5, 6)\n",
    "    # 標本空間\n",
    "    s = die_sides**max_rolls\n",
    "    # Find the event set\n",
    "    if max_rolls > 1:\n",
    "        success_rolls = []\n",
    "        for elem in s:\n",
    "            if sum(elem) >= target_score:\n",
    "                success_rolls.append(elem)\n",
    "    else:\n",
    "        if target_score > 6:\n",
    "            success_rolls = []\n",
    "        else:\n",
    "            success_rolls = []\n",
    "            for roll in die_sides:\n",
    "                success_rolls.append(roll)\n",
    "    e = FiniteSet(*success_rolls)\n",
    "    # 目標点数に達する確率の計算\n",
    "    return len(e)/len(s)\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    target_score = 25\n",
    "    max_roll = 4\n",
    "    p = find_prob(target_score, max_roll)\n",
    "    print('Probability: {0:.5f}'.format(p))\n",
    "    \n",
    "    target_score = 25\n",
    "    max_roll = 5\n",
    "    p = find_prob(target_score, max_roll)\n",
    "    print('Probability: {0:.5f}'.format(p))    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 非一様乱数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "50\n"
     ]
    }
   ],
   "source": [
    "import random\n",
    "\n",
    "\n",
    "def toss():\n",
    "    # 0 -> Heads, 1 -> Tails\n",
    "    if random.random() < 2/3:\n",
    "        return 0\n",
    "    else:\n",
    "        return 1\n",
    "\n",
    "\n",
    "'''\n",
    "各種紙幣を異なる確率で払う空想ATMのシミュレーション\n",
    "'''\n",
    "import random\n",
    "\n",
    "\n",
    "def get_index(probability):\n",
    "    c_probability = 0\n",
    "    sum_probability = []\n",
    "    for p in probability:\n",
    "        c_probability += p\n",
    "        sum_probability.append(c_probability)\n",
    "        r = random.random()\n",
    "        for index, sp in enumerate(sum_probability):\n",
    "            if r <= sp:\n",
    "                return index\n",
    "        return len(probability) - 1\n",
    "\n",
    "\n",
    "def dispens():\n",
    "    dollar_bills = [5, 10, 20, 50]\n",
    "    probability = [1/6, 1/6, 1/3, 1/3]\n",
    "    bill_index = get_index(probability)\n",
    "    return dollar_bills[bill_index]\n",
    "\n",
    "  \n",
    "if __name__ == '__main__':\n",
    "    print(dispens())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "ename": "ModuleNotFoundError",
     "evalue": "No module named 'polar'",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mModuleNotFoundError\u001b[0m                       Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-99af4738a9b0>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0msys\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;32mimport\u001b[0m \u001b[0mpolar\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      3\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mnumpy\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m31\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'polar'"
     ],
     "output_type": "error"
    }
   ],
   "source": [
    "import sys\n",
    "import polar\n",
    "import numpy as np\n",
    "\n",
    "r = np.linspace(0, 1, 31)\n",
    "theta = np.linspace(0, 4.0 * np.pi, 31)\n",
    "x, y = polar.get_xy(r, theta)"
   ]
  }
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