czsheets/Riddler_Express_9_7_18.ipynb

## blog.md

      
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              blog.md
            
          
    This week's Riddler Express draws from the game show Who Wants to Be a Millionaire:
You are a contestant on “Who Wants to Be a Riddler Millionaire.” You have already made it to a late round: You could walk away right now with $250,000. But there are two potential questions still to go that you can try to answer. You could earn $500,000 if you get one right and then walk away, or $1 million if you nail them both. If you attempt any answer and miss, you go home with $10,000.
The 50/50: The host reduces the four possible answers to two; one of them is the correct one and the other is randomly chosen from among the other three answers.
Ask the Audience: The studio audience submits their own guesses.
You know historically that the correct answer will be chosen by the plurality 50 percent of the time; while 30 percent of the time the right answer finishes second; 15 percent third; and 5 percent last. Additionally, if there are only two answers available to the audience, they pick the correct one more often 65 percent of the time.
Aside from leaving with the $250,000 you already have, there are four options that I can think of here:

Option 1: use both hints on the first question, then quit
Option 2: use both hints on the first question, guess on the second
Option 3: guess on the first question, and use both hints on the second
Option 4: use one hint on each question

I simulated the outcome to approximate the values for each of these scenarios, and Option 1 ($328,219) is the only one that provides a meaningful improvement in your expected value. Using one hint on each question (Option 4) increased the expected value slightly ($257,631) while Options 2 ($193,251) and 3 ($170600) both have an expected value that is less than just walking away. Of course, as a very smart friend pointed out, expected value works for the game show, who - if they last long enough - has many contestants (similar to a casino). As a player you only get one chance at this, so you have to consider whether the increase in expected value of ~ $80,000 is worth the risk of losing almost a quarter million dollars. Results are in the tables and chart below, and the Jupyter Python code is on Github.


Option
Expected value


Option 1
328,219


Option 2
193,251


Option 3
170,600


Option 4
257,631


## gistlog.yml
published: true

## Riddler_Express_9_7_18.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# You are a contestant on “Who Wants to Be a Riddler Millionaire.” \n",
    "# You have already made it to a late round: You could walk away right now with $250,000. \n",
    "# But there are two potential questions still to go that you can try to answer. \n",
    "# You could earn $500,000 if you get one right and then walk away, or $1 million if you nail them both. \n",
    "# If you attempt any answer and miss, you go home with $10,000.\n",
    "\n",
    "# The 50/50: The host reduces the four possible answers to two; \n",
    "# one of them is the correct one and the other is randomly chosen from among the other three answers.\n",
    "\n",
    "# Ask the Audience: The studio audience submits their own guesses. \n",
    "# You know historically that the correct answer will be chosen by the plurality 50 percent of the time; \n",
    "# while 30 percent of the time the right answer finishes second; 15 percent third; and 5 percent last. \n",
    "# Additionally, if there are only two answers available to the audience, they pick the correct one more often 65 percent of the time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "from matplotlib import pyplot as plt\n",
    "import pandas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Option 1: the expected value of using both hints on the first question and then quitting is: $328219\n"
     ]
    }
   ],
   "source": [
    "both_first_question = []\n",
    "for chance in range(1000000):\n",
    "    chance = numpy.random.choice([500000,10000],1,p = [.65, .35])\n",
    "    both_first_question.extend(chance)\n",
    "print(\"Option 1: the expected value of using both hints on the first question and then quitting is: $\" +\n",
    "      str(int(numpy.mean(both_first_question))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Option 2: the expected value of using both hints on the first question and then guessing on the second is: $193251\n"
     ]
    }
   ],
   "source": [
    "keep_going = []\n",
    "for chance in range(1000000):\n",
    "    chance = numpy.random.choice([0,1],1, p = [.35, .65])\n",
    "    if chance == 0:\n",
    "        keep_going.append(10000)\n",
    "    else:\n",
    "        next_chance = numpy.random.choice([1000000,10000],1,p = [.25,.75])\n",
    "    keep_going.extend(next_chance)\n",
    "print(\"Option 2: the expected value of using both hints on the first question and then guessing on the second is: $\" +\n",
    "      str(int(numpy.mean(keep_going))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Option 3: the expected value of guessing on the first question and then using both hints on the second question is: $170600\n"
     ]
    }
   ],
   "source": [
    "both_second_question = []\n",
    "for chance in range(1000000):\n",
    "    chance = numpy.random.choice([0,1],1,p = [.75,.25])\n",
    "    if chance == 0:\n",
    "        both_second_question.append(10000)\n",
    "    else:\n",
    "        next_chance = numpy.random.choice([1000000,10000],1,p = [.65,.35])\n",
    "        both_second_question.append(next_chance)\n",
    "print(\"Option 3: the expected value of guessing on the first question and then using both hints on the second question is: $\" +\n",
    "      str(int(numpy.mean(both_second_question))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Option 4: the expected value of using one hint on each question is: $257631\n"
     ]
    }
   ],
   "source": [
    "one_each = []\n",
    "for chance in range(1000000):\n",
    "    chance = numpy.random.choice([0,1],1)\n",
    "    if chance == 0:\n",
    "        one_each.append(10000)\n",
    "    else:\n",
    "        chance = numpy.random.choice([0,1],1)\n",
    "        if chance == 0:\n",
    "            one_each.append(10000)\n",
    "        else:\n",
    "            one_each.append(1000000)\n",
    "print(\"Option 4: the expected value of using one hint on each question is: $\" +\n",
    "      str(int(numpy.mean(one_each))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = {\"Option1\" : numpy.mean(both_first_question), \"Option2\": numpy.mean(keep_going),\n",
    "              \"Option3\": numpy.mean(one_each), \"Option4\": numpy.mean(both_second_question)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
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k6e+B2yjNZfxDRPy2wkMyK5TvNMwKImkQcBnwc+AV4OuSelR2VGbFcmiYFecR\n4HsR8Q1Kr9SuAp6u7JDMiuXHU2YFkdQzIjY3qh0eESsrNSazovlOw6yNJF0JEBGbJZ3d6PAFHT8i\ns/bj0DBru/Fl21c3OjamIwdi1t4cGmZtp2a2m9o32605NMzaLprZbmrfbLfmiXCzNpK0jdL3MgR0\nB95rOARURcQ+lRqbWdEcGmZmls2Pp8zMLJtDw8zMsjk0zMwsm0PDzMyy/X9zcEoLJIcwFAAAAABJ\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d9173ec048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pandas.DataFrame(results, index=['Expected\\nValue']).plot(kind='bar')\n",
    "plt.show()"
   ]
  }
 ],
 "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",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
Option	Expected value
Option 1	328,219
Option 2	193,251
Option 3	170,600
Option 4	257,631
	{
	"cells": [
	{
	"cell_type": "code",
	"execution_count": 1,
	"metadata": {
	"collapsed": true
	},
	"outputs": [],
	"source": [
	"# You are a contestant on “Who Wants to Be a Riddler Millionaire.” \n",
	"# You have already made it to a late round: You could walk away right now with $250,000. \n",
	"# But there are two potential questions still to go that you can try to answer. \n",
	"# You could earn $500,000 if you get one right and then walk away, or $1 million if you nail them both. \n",
	"# If you attempt any answer and miss, you go home with $10,000.\n",
	"\n",
	"# The 50/50: The host reduces the four possible answers to two; \n",
	"# one of them is the correct one and the other is randomly chosen from among the other three answers.\n",
	"\n",
	"# Ask the Audience: The studio audience submits their own guesses. \n",
	"# You know historically that the correct answer will be chosen by the plurality 50 percent of the time; \n",
	"# while 30 percent of the time the right answer finishes second; 15 percent third; and 5 percent last. \n",
	"# Additionally, if there are only two answers available to the audience, they pick the correct one more often 65 percent of the time."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 67,
	"metadata": {
	"collapsed": true
	},
	"outputs": [],
	"source": [
	"import numpy\n",
	"from matplotlib import pyplot as plt\n",
	"import pandas"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 116,
	"metadata": {},
	"outputs": [
	{
	"name": "stdout",
	"output_type": "stream",
	"text": [
	"Option 1: the expected value of using both hints on the first question and then quitting is: $328219\n"
	]
	}
	],
	"source": [
	"both_first_question = []\n",
	"for chance in range(1000000):\n",
	" chance = numpy.random.choice([500000,10000],1,p = [.65, .35])\n",
	" both_first_question.extend(chance)\n",
	"print(\"Option 1: the expected value of using both hints on the first question and then quitting is: $\" +\n",
	" str(int(numpy.mean(both_first_question))))"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 117,
	"metadata": {},
	"outputs": [
	{
	"name": "stdout",
	"output_type": "stream",
	"text": [
	"Option 2: the expected value of using both hints on the first question and then guessing on the second is: $193251\n"
	]
	}
	],
	"source": [
	"keep_going = []\n",
	"for chance in range(1000000):\n",
	" chance = numpy.random.choice([0,1],1, p = [.35, .65])\n",
	" if chance == 0:\n",
	" keep_going.append(10000)\n",
	" else:\n",
	" next_chance = numpy.random.choice([1000000,10000],1,p = [.25,.75])\n",
	" keep_going.extend(next_chance)\n",
	"print(\"Option 2: the expected value of using both hints on the first question and then guessing on the second is: $\" +\n",
	" str(int(numpy.mean(keep_going))))"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 118,
	"metadata": {},
	"outputs": [
	{
	"name": "stdout",
	"output_type": "stream",
	"text": [
	"Option 3: the expected value of guessing on the first question and then using both hints on the second question is: $170600\n"
	]
	}
	],
	"source": [
	"both_second_question = []\n",
	"for chance in range(1000000):\n",
	" chance = numpy.random.choice([0,1],1,p = [.75,.25])\n",
	" if chance == 0:\n",
	" both_second_question.append(10000)\n",
	" else:\n",
	" next_chance = numpy.random.choice([1000000,10000],1,p = [.65,.35])\n",
	" both_second_question.append(next_chance)\n",
	"print(\"Option 3: the expected value of guessing on the first question and then using both hints on the second question is: $\" +\n",
	" str(int(numpy.mean(both_second_question))))"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 119,
	"metadata": {},
	"outputs": [
	{
	"name": "stdout",
	"output_type": "stream",
	"text": [
	"Option 4: the expected value of using one hint on each question is: $257631\n"
	]
	}
	],
	"source": [
	"one_each = []\n",
	"for chance in range(1000000):\n",
	" chance = numpy.random.choice([0,1],1)\n",
	" if chance == 0:\n",
	" one_each.append(10000)\n",
	" else:\n",
	" chance = numpy.random.choice([0,1],1)\n",
	" if chance == 0:\n",
	" one_each.append(10000)\n",
	" else:\n",
	" one_each.append(1000000)\n",
	"print(\"Option 4: the expected value of using one hint on each question is: $\" +\n",
	" str(int(numpy.mean(one_each))))"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 120,
	"metadata": {},
	"outputs": [],
	"source": [
	"results = {\"Option1\" : numpy.mean(both_first_question), \"Option2\": numpy.mean(keep_going),\n",
	" \"Option3\": numpy.mean(one_each), \"Option4\": numpy.mean(both_second_question)}"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 121,
	"metadata": {},
	"outputs": [
	{
	"data": {
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X43FNklQrqXb9+vW7cklmZrYLskND0n7Az4HLI2Jz+bF05xAFjy1bRNwRETURUdOnT59K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\nRU5ErkJggg==\n",
	"text/plain": [
	"<matplotlib.figure.Figure at 0x1d9173ec048>"
	]
	},
	"metadata": {},
	"output_type": "display_data"
	}
	],
	"source": [
	"pandas.DataFrame(results, index=['Expected\\nValue']).plot(kind='bar')\n",
	"plt.show()"
	]
	}
	],
	"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",
	"version": "3.6.1"
	}
	},
	"nbformat": 4,
	"nbformat_minor": 2
	}