정사면체

기출문제(정사면체).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [서울과학고 기통 기출문제]\n",
    "Q. 각 면에 $1,2,3,6$의 숫자가 하나씩 적혀 있는 정사면체가 있다. 이 정사면체를 세 번 던져서 얻은 밑면에 적힌 숫자를\n",
    "나온 순서대로 $x,y,z$라 하자. \n",
    "다음 물음에 답하시오.\n",
    "- (1) $xyz$가 6의 배수가 될 확률을 구하시오\n",
    "- (2) $xy+yz+zx$가 3의 배수가 될 확률을 구하시오"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 시뮬레이션 해보기"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "tetr = [1,2,3,6]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.choice(tetr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "random.choices?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2, 2, 3]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.choices(tetr,k=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "def myt(mytetr,mycount):\n",
    "    \"\"\"xyz가 6의배수일 확률\"\"\"\n",
    "    su =0\n",
    "    for i in range(mycount):\n",
    "        if random.choice(tetr)*random.choice(tetr)*random.choice(tetr) % 6 ==0:\n",
    "           # print(mytetr[:3])\n",
    "            su =su+1\n",
    "    return su/mycount\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7671"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "myt(tetr,10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def myt2(mytetr,mycount):\n",
    "    \"\"\"xy+yz+zx가 3의배수일 확률\"\"\"\n",
    "    su =0\n",
    "    \n",
    "    for i in range(mycount):\n",
    "        x,y,z = random.choices(mytetr,k=3)\n",
    "        if (x*y+y*z+z*x) % 3 ==0:\n",
    "           # print(mytetr[:3])\n",
    "            su =su+1\n",
    "    return su/mycount"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x,y,z =[1,2,3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5279"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "myt2(tetr,10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 실제 계산해보기"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import itertools\n",
    "import fractions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "itertools.product?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "fractions.Fraction?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 (1, 1, 1)\n",
      "2 (1, 1, 2)\n",
      "3 (1, 1, 3)\n",
      "4 (1, 1, 6)\n",
      "5 (1, 2, 1)\n",
      "6 (1, 2, 2)\n",
      "7 (1, 2, 3)\n",
      "8 (1, 2, 6)\n",
      "9 (1, 3, 1)\n",
      "10 (1, 3, 2)\n",
      "11 (1, 3, 3)\n",
      "12 (1, 3, 6)\n",
      "13 (1, 6, 1)\n",
      "14 (1, 6, 2)\n",
      "15 (1, 6, 3)\n",
      "16 (1, 6, 6)\n",
      "17 (2, 1, 1)\n",
      "18 (2, 1, 2)\n",
      "19 (2, 1, 3)\n",
      "20 (2, 1, 6)\n",
      "21 (2, 2, 1)\n",
      "22 (2, 2, 2)\n",
      "23 (2, 2, 3)\n",
      "24 (2, 2, 6)\n",
      "25 (2, 3, 1)\n",
      "26 (2, 3, 2)\n",
      "27 (2, 3, 3)\n",
      "28 (2, 3, 6)\n",
      "29 (2, 6, 1)\n",
      "30 (2, 6, 2)\n",
      "31 (2, 6, 3)\n",
      "32 (2, 6, 6)\n",
      "33 (3, 1, 1)\n",
      "34 (3, 1, 2)\n",
      "35 (3, 1, 3)\n",
      "36 (3, 1, 6)\n",
      "37 (3, 2, 1)\n",
      "38 (3, 2, 2)\n",
      "39 (3, 2, 3)\n",
      "40 (3, 2, 6)\n",
      "41 (3, 3, 1)\n",
      "42 (3, 3, 2)\n",
      "43 (3, 3, 3)\n",
      "44 (3, 3, 6)\n",
      "45 (3, 6, 1)\n",
      "46 (3, 6, 2)\n",
      "47 (3, 6, 3)\n",
      "48 (3, 6, 6)\n",
      "49 (6, 1, 1)\n",
      "50 (6, 1, 2)\n",
      "51 (6, 1, 3)\n",
      "52 (6, 1, 6)\n",
      "53 (6, 2, 1)\n",
      "54 (6, 2, 2)\n",
      "55 (6, 2, 3)\n",
      "56 (6, 2, 6)\n",
      "57 (6, 3, 1)\n",
      "58 (6, 3, 2)\n",
      "59 (6, 3, 3)\n",
      "60 (6, 3, 6)\n",
      "61 (6, 6, 1)\n",
      "62 (6, 6, 2)\n",
      "63 (6, 6, 3)\n",
      "64 (6, 6, 6)\n"
     ]
    }
   ],
   "source": [
    "for j,i in enumerate(itertools.product([1,2,3,6],repeat=3),1):\n",
    "    print(j,i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "첫번째 답은 49/64, 근사 0.765625\n",
      "두번째 답은 17/32, 근사 0.53125\n"
     ]
    }
   ],
   "source": [
    "msu,msu2 =0,0\n",
    "for i in itertools.product([1,2,3,6],repeat=3):\n",
    "    x,y,z = i\n",
    "    if (x*y*z)% 6 == 0:\n",
    "        msu = msu+1\n",
    "    if (x*y+y*z+z*x)% 3 ==0:\n",
    "        msu2 = msu2+1\n",
    "print(\"첫번째 답은 {}, 근사 {}\".format(fractions.Fraction(msu/64), msu/64))\n",
    "print(\"두번째 답은 {}, 근사 {}\".format(fractions.Fraction(msu2/64), msu2/64))\n"
   ]
  }
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