Monte Carlo Simulation of the Monty Hall Problem: Does The Math Actually Work?
Introduction
If you ever took a stats class back in high school or college, then you were probably told about the infamous Monty Hall problem. The premise is very simple:
The contestant is presented with 3 doors. Behind one of the doors, there is a car, and behind the other two, there is a goat.
The contestant picks a door, then the host reveals one of the other two doors.
The contestant then has a choice: should she switch doors or stick with what she picked first?
You can play the game in this link, try it out for yourself!
You can find the full article here.
Now here is the full code (in Python)
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import random
%matplotlib inline
def generate_doors():
door1 = random.randint(0, 1)
if door1 == 1:
door2 = 0
door3 = 0
door_with_car = 0
elif door1 == 0:
door2 = random.randint(0, 1)
if door2 == 1:
door3 = 0
door_with_car = 1
else:
door3 = 1
door_with_car = 2
all_doors = [door1, door2, door3]
return all_doors, door_with_car
def reveal_door(door_with_car, player_choice):
lst = [0, 1, 2]
try:
lst.remove(door_with_car)
lst.remove(player_choice)
except:
pass
return random.choice(lst)
score = {'player_keep': 0, 'player_switch': 0}
# Setting up the game
all_doors, door_with_car = generate_doors()
print("We start with 3 doors with the door of '1' having the car:", all_doors)
# Player makes a choice of door
player_choice = random.randint(0, 2)
print("Player chooses door #", player_choice + 1)
# Host reveals the door
revealed_door = reveal_door(door_with_car, player_choice)
print("Host reveals door #", revealed_door + 1)
# Remove revealed door from list
del all_doors[revealed_door]
print("Now there are only 2 doors left to choose:", all_doors)
# Player chooses between keeping or switching
player_keep = player_choice
player_switch = [x for x in all_doors if x != player_choice][0]
# Decide who's right
if player_keep == door_with_car:
score['player_keep'] += 1
print("If player chooses to keep, he wins!")
else:
score['player_switch'] += 1
print("If player chooses to switch, he wins!")
score = {'Keep Choice': 0, 'Switch Choice': 0}
for i in range(10000):
# Setting up the game
all_doors, door_with_car = generate_doors()
# Player makes a choice of door
player_choice = random.randint(0, 2)
# Host reveals the door
revealed_door = reveal_door(door_with_car, player_choice)
# Remove revealed door from list
del all_doors[revealed_door]
# Player chooses between keeping or switching
player_keep = player_choice
player_switch = [x for x in all_doors if x != player_choice][0]
# Decide who's right
if player_keep == door_with_car:
score['Keep Choice'] += 1
else:
score['Switch Choice'] += 1
score
pd.DataFrame.from_dict(score, orient='index').plot.bar(legend=False, title="Count of Wins for Each Strategy", figsize=(7, 4));