{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 2: Conditional probability\n", " \n", "This Jupyter notebook is the Python equivalent of the R code in section 2.10 R, pp. 72 - 74, [Introduction to Probability, 1st Edition](https://www.crcpress.com/Introduction-to-Probability/Blitzstein-Hwang/p/book/9781466575578), Blitzstein & Hwang.\n", "\n", "----" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Simulating the frequentist interpretation \n", "\n", "Recall that the frequentist interpretation of conditional probability based on a large number `n` of repetitions of an experiment is $P(A|B) ≈ n_{AB}/n_{B}$, where $n_{AB}$ is the number of times that $A \\cap B$ occurs and $n_{B}$ is the number of times that $B$ occurs. Let's try this out by simulation, and verify the results of Example 2.2.5. So let's use [`numpy.random.choice`](https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.random.choice.html) to simulate `n` families, each with two children." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "child1:\n", "[2 1 1 ... 1 2 1]\n", "\n", "child2:\n", "[2 2 2 ... 2 2 1]\n", "\n" ] } ], "source": [ "np.random.seed(34)\n", "\n", "n = 10**5\n", "child1 = np.random.choice([1,2], n, replace=True) \n", "child2 = np.random.choice([1,2], n, replace=True) \n", "\n", "print('child1:\\n{}\\n'.format(child1))\n", "\n", "print('child2:\\n{}\\n'.format(child2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here `child1` is a NumPy `array` of length `n`, where each element is a 1 or a 2. Letting 1 stand for \"girl\" and 2 stand for \"boy\", this `array` represents the gender of the elder child in each of the `n` families. Similarly, `child2` represents the gender of the younger child in each family. \n", "\n", "Alternatively, we could have used" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array(['boy', 'boy', 'boy', ..., 'boy', 'boy', 'boy'], dtype='5 runs of the Monty Hall game. To simplify notation, assume the contestant always chooses door 1. Then we can generate a vector specifying which door has the car for each repetition:\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The never-switch strategy has success rate 0.331\n" ] } ], "source": [ "np.random.seed(55)\n", "\n", "n = 10**5\n", "cardoor = np.random.choice([1,2,3] , n, replace=True)\n", "\n", "print('The never-switch strategy has success rate {:.3F}'.format(np.sum(cardoor==1) / n))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At this point we could generate the vector specifying which doors Monty opens, but that's unnecessary since the never-switch strategy succeeds if and only if door 1 has the car! So the fraction of times when the never-switch strategy succeeds is `numpy.sum(cardoor==1)/n`, which was 0.331in our simulation. This is very close to 1/3.\n", "\n", "What if we want to play the Monty Hall game interactively? We can do this by programming a Python class that would let us play interactively or let us run a simulation across many trials." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "class Monty():\n", " \n", " def __init__(self):\n", " \"\"\" Object creation function. \"\"\"\n", " self.state = 0\n", " self.doors = np.array([1, 2, 3])\n", " self.prepare_game()\n", "\n", " def get_success_rate(self):\n", " \"\"\" Return the rate of success in this series of plays: num. wins / num. plays. \"\"\"\n", " if self.num_plays > 0:\n", " return 1.0*self.num_wins / self.num_plays\n", " else:\n", " return 0.0\n", "\n", " def prepare_game(self):\n", " \"\"\" Prepare initial values for game play, and randonly choose the door with the car. \"\"\"\n", " self.num_plays = 0\n", " self.num_wins = 0\n", " self.cardoor = np.random.choice(self.doors)\n", " self.players_choice = None\n", " self.montys_choice = None\n", " \n", " def choose_door(self, door):\n", " \"\"\" Player chooses a door at state 0. Monty will choose a remaining door to reveal a goat. \"\"\"\n", " self.state = 1\n", " self.players_choice = door\n", " self.montys_choice = np.random.choice(self.doors[(self.doors!=self.players_choice) & (self.doors!=self.cardoor)])\n", " \n", " def switch_door(self, do_switch):\n", " \"\"\" Player has the option to switch from the door she has chosen to the remaining unopened door. \n", " \n", " If the door the player has selected is the same as the cardoor, then num. of wins is incremented.\n", " \n", " Finally, number of plays will be incremented.\n", " \"\"\"\n", " self.state = 2\n", " if do_switch:\n", " self.players_choice = self.doors[(self.doors!=self.players_choice) & (self.doors!=self.montys_choice)][0]\n", " if self.players_choice == self.cardoor:\n", " self.num_wins += 1\n", " self.num_plays += 1\n", " \n", " def continue_play(self):\n", " \"\"\" Player opts to continue playing in this series. \n", " \n", " The game is returned to state 0, but the counters for num. wins and num. plays\n", " will be kept intact and running.\n", " \n", " A new cardoor is randomly chosen.\n", " \"\"\"\n", " self.state = 0\n", " self.cardoor = np.random.choice(self.doors)\n", " self.players_choice = None\n", " self.montys_choice = None\n", " \n", " def reset(self):\n", " \"\"\" The entire game state is returned to its initial state. \n", " \n", " All counters and variable holdling state are re-initialized.\n", " \"\"\"\n", " self.state = 0\n", " self.prepare_game()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In brief:\n", "* The `Monty` class represents a simple state model for the game.\n", "* When an instance of the `Monty` game is created, game state-holding variables are initialized and a `cardoor` randomly chosen.\n", "* After the player initially picks a door, `Monty` will choose a remaining door that does not have car behind it.\n", "* The player can then choose to switch to the other, remaining unopened door, or stick with her initial choice.\n", "* `Monty` will then see if the player wins or not, and updates the state-holding variables for num. wins and num. plays.\n", "* The player can continue playing, or stop and reset the game to its original state.\n", "\n", "### As a short simulation program\n", "\n", "Here is an example showing how to use the `Monty` class above to run a simulation to see how often the switching strategy succeeds." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "In 100000 trials, the switching strategy won 66730 times.\n", "Success rate is 0.667\n" ] } ], "source": [ "np.random.seed(89)\n", "\n", "trials = 10**5\n", "\n", "game = Monty()\n", "for _ in range(trials):\n", " game.choose_door(np.random.choice([1,2,3]))\n", " game.switch_door(True)\n", " game.continue_play()\n", "\n", "print('In {} trials, the switching strategy won {} times.'.format(game.num_plays, game.num_wins))\n", "print('Success rate is {:.3f}'.format(game.get_success_rate()))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### As an interactive widget in this Jupyter notebook\n", "\n", "Optionally, the `Monty` Python class above can also be used as an engine to power an interactive widget that lets you play the three-door game _in the browser_ using [`ipywidgets` ](https://ipywidgets.readthedocs.io/en/stable/user_guide.html).\n", "\n", "To run the interactive widget, make sure you have the `ipywidgets` package installed (v7.4.2 or greater).\n", "\n", "To install with the `conda` package manager, execute the following command:\n", "\n", " conda install ipywidgets\n", "\n", "To install with the `pip` package manager, execute the following command:\n", "\n", " pip install ipywidgets" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "from ipywidgets import Box, Button, ButtonStyle, FloatText, GridBox, IntText, Label, Layout, HBox\n", "from IPython.display import display" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The doors in the game are represented by [`ipywidgets.Button`](https://ipywidgets.readthedocs.io/en/stable/examples/Widget%20List.html#Button)." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "door1 = Button(description='Door 1', layout=Layout(flex='1 1 auto', width='auto'))\n", "door2 = Button(description='Door 2', layout=door1.layout)\n", "door3 = Button(description='Door 3', layout=door1.layout)\n", "doors_arr = [door1, door2, door3]\n", "doors = Box(doors_arr, layout=Layout(width='auto', grid_area='doors'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "State-holding variables in the `Monty` object are displayed using [`ipywidgets.IntText`](https://ipywidgets.readthedocs.io/en/stable/examples/Widget%20List.html#IntText) (for the `num_wins` and `num_plays`); and [`ipywidgets.FloatText`](https://ipywidgets.readthedocs.io/en/stable/examples/Widget%20List.html#FloatText) (for the success rate)." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "label1 = Label(value='number of plays', layout=Layout(width='auto', grid_area='label1'))\n", "text1 = IntText(disabled=True, layout=Layout(width='auto', grid_area='text1'))\n", "\n", "label2 = Label(value='number of wins', layout=Layout(width='auto', grid_area='label2'))\n", "text2 = IntText(disabled=True, layout=Layout(width='auto', grid_area='text2'))\n", "\n", "label3 = Label(value='success rate', layout=Layout(width='auto', grid_area='label3'))\n", "text3 = FloatText(disabled=True, layout=Layout(width='auto', grid_area='text3'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "[`ipywidgets.Label`](https://ipywidgets.readthedocs.io/en/stable/examples/Widget%20List.html#Label) is used to display the title and descriptive text in the game widget." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "banner = Box([Label(value='Interactive widget: Monty Hall problem', \n", " layout=Layout(width='50%'))], \n", " layout=Layout(width='auto', justify_content='center', grid_area='banner'))\n", "\n", "status = Label(value='Pick a door...', layout=Layout(width='auto', grid_area='status'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Buttons allowing for further user actions are located at the bottom of the widget.\n", "\n", "* The `reveal` button is used to show what's behind all of the doors after the player makes her final choice.\n", "* After the player completes a round of play, she can click the `continue` button to keep counting game state (num. wins and num. plays)\n", "* The `reset` button lets the player return the game to its original state after completing a round of play." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "button_layout = Layout(flex='1 1 auto', width='auto')\n", "reveal = Button(description='reveal', tooltip='open selected door', layout=button_layout, disabled=True)\n", "contin = Button(description='continue', tooltip='continue play', layout=button_layout, disabled=True)\n", "reset = Button(description='reset', tooltip='reset game', layout=button_layout, disabled=True)\n", "actions = Box([reveal, contin, reset], layout=Layout(width='auto', grid_area='actions'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "[`ipywidgets.GridBox`](https://ipywidgets.readthedocs.io/en/stable/examples/Widget%20Styling.html#The-Grid-layout) helps us lay out the user interface elements for the `Monty` game widget." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "ui = GridBox(children=[banner, doors, label1, text1, label2, text2, label3, text3, status, actions],\n", " layout=Layout(\n", " width='50%',\n", " grid_template_rows='auto auto auto auto auto auto auto',\n", " grid_template_columns='25% 25% 25% 25%',\n", " grid_template_areas='''\n", " \"banner banner banner banner\"\n", " \"doors doors doors doors\"\n", " \"label1 label1 text1 text1\"\n", " \"label2 label2 text2 text2\"\n", " \"label3 label3 text3 text3\"\n", " \"status status status status\"\n", " \". . actions actions\"\n", " '''\n", " )\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We lastly create some functions to connect the widget to the `Monty` game object. These functions adapt player action [events](https://ipywidgets.readthedocs.io/en/stable/examples/Widget%20Events.html#Example) to state changes in the `Monty` object, and then update the widget user interface accordingly." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "de1cbd6b692f4f6abaf2bf8f2a4e9b8f", "version_major": 2, "version_minor": 0 }, "text/plain": [ "GridBox(children=(Box(children=(Label(value='Interactive widget: Monty Hall problem', layout=Layout(width='50%…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "uigame = Monty()\n", "\n", "def reset_ui(disable_reset=True):\n", " \"\"\" Return widget elements to their initial state.\n", " Do not disable the reset button in the case of continue.\n", " \"\"\"\n", " for i,d in enumerate(doors_arr):\n", " d.description = 'Door {}'.format(i+1)\n", " d.disabled = False\n", " d.icon = ''\n", " d.button_style = ''\n", " \n", " reveal.disabled = True\n", " contin.disabled = True\n", " reset.disabled = disable_reset\n", " \n", "def update_status(new_status):\n", " \"\"\" Update the widget text fields for displaying present game status. \"\"\"\n", " text1.value = uigame.num_plays\n", " text2.value = uigame.num_wins\n", " text3.value = uigame.get_success_rate()\n", " status.value = new_status\n", " \n", "def update_ui_reveal():\n", " \"\"\" Helper function to update the widget after the player clicks the reveal button. \"\"\"\n", " if uigame.players_choice == uigame.cardoor:\n", " new_status = 'You win! Continue playing?'\n", " else:\n", " new_status = 'Sorry, you lose. Continue playing?'\n", "\n", " for i,d in enumerate(doors_arr):\n", " d.disabled = True\n", " if uigame.cardoor == i+1:\n", " d.description = 'car'\n", " else:\n", " d.description = 'goat'\n", " if uigame.players_choice == i+1:\n", " if uigame.players_choice == uigame.cardoor:\n", " d.button_style = 'success' \n", " d.icon = 'check'\n", " else:\n", " d.button_style = 'danger' \n", " d.icon = 'times'\n", " update_status(new_status)\n", " reveal.disabled = True\n", " contin.disabled = False\n", " reset.disabled = False \n", " \n", "def on_button_clicked(b): \n", " \"\"\" Event-handling function that maps button click events in the widget \n", " to corresponding functions in Monty, and updates the user interface\n", " according to the present game state.\n", " \"\"\"\n", " if uigame.state == 0:\n", " if b.description in ['Door 1', 'Door 2', 'Door 3']:\n", " c = int(b.description.split()[1])\n", " uigame.choose_door(c)\n", "\n", " b.disabled = True\n", " b.button_style = 'info'\n", "\n", " m = doors_arr[uigame.montys_choice-1]\n", " m.disabled = True\n", " m.description = 'goat'\n", "\n", " unopened = uigame.doors[(uigame.doors != uigame.players_choice) &\n", " (uigame.doors != uigame.montys_choice)][0]\n", " status.value = 'Monty reveals a goat behind Door {}. Click Door {} to switch, or \\'reveal\\' Door {}.' \\\n", " .format(uigame.montys_choice, unopened, uigame.players_choice)\n", "\n", " reveal.disabled = False\n", " reset.disabled = False\n", " elif b.description == 'reset':\n", " uigame.reset()\n", " reset_ui()\n", " update_status('Pick a door...') \n", " elif uigame.state == 1:\n", " if b.description in ['Door 1', 'Door 2', 'Door 3']:\n", " prev_choice = uigame.players_choice\n", " uigame.switch_door(True)\n", " \n", " pb = doors_arr[prev_choice-1]\n", " pb.icon = ''\n", " pb.button_style = ''\n", " \n", " b.disabled = True\n", " b.button_style = 'info'\n", " \n", " status.value = 'Now click \\'reveal\\' to see what\\'s behind Door {}.'.format(uigame.players_choice)\n", " \n", " elif b.description == 'reset':\n", " uigame.reset()\n", " reset_ui()\n", " update_status('Pick a door...')\n", " elif b.description == 'reveal':\n", " uigame.switch_door(False)\n", " update_ui_reveal()\n", " elif uigame.state == 2:\n", " if b.description == 'reveal':\n", " update_ui_reveal() \n", " else: \n", " if b.description == 'continue':\n", " uigame.continue_play()\n", " reset_ui(False)\n", " update_status('Pick a door once more...')\n", " elif b.description == 'reset':\n", " uigame.reset()\n", " reset_ui()\n", " update_status('Pick a door...')\n", "\n", "# hook up all buttons to our event-handling function\n", "door1.on_click(on_button_clicked)\n", "door2.on_click(on_button_clicked)\n", "door3.on_click(on_button_clicked)\n", "reveal.on_click(on_button_clicked)\n", "contin.on_click(on_button_clicked)\n", "reset.on_click(on_button_clicked)\n", "\n", "display(ui)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How to play:\n", "* Click a door to select.\n", "* Monty will select a remaining door and open to reveal a goat.\n", "* Click the `reveal` button to open your selected door.\n", "* Or click the remaining unopened to switch doors, and then click `reveal`.\n", "* Click the `continue` button to keep playing.\n", "* You may click the `reset` button at any time to return the game back to its initial state." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "----\n", "\n", "© Blitzstein, Joseph K.; Hwang, Jessica. Introduction to Probability (Chapman & Hall/CRC Texts in Statistical Science)." ] } ], "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": "3.7.1" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": { "0023fdeb843146bfb8c334ab446160de": { "model_module": "@jupyter-widgets/controls", "model_module_version": "1.4.0", "model_name": "ButtonModel", "state": { "description": "goat", "disabled": true, "layout": "IPY_MODEL_f0e464f3ba0646b3af0a8700ad12c39b", "style": "IPY_MODEL_f5d85ac86acf44368b18a307ffbebaec" } }, "005ab726996d42c78d2629c2b4cad00b": { "model_module": "@jupyter-widgets/controls", "model_module_version": "1.4.0", "model_name": "DescriptionStyleModel", "state": { "description_width": "" } }, 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