使用 Pygame 模拟蒙蒂霍尔问题

简述

蒙蒂霍尔问题（Monty Hall problem）是一个概率谜题，由美国主持人蒙蒂·霍尔（Monty Hall）所提出，题目如下：

有三扇门，其中一扇门后面有一辆汽车，另外两扇门后面则各有一只山羊。参赛者选择一扇门，蒙蒂打开剩下两扇门的其中一扇，露出其中一只山羊。然后，蒙蒂问参赛者是否要换另一扇门。问题是：换另一扇门会否增加参赛者赢得汽车的机会？

答案是：换另一扇门可以让参赛者的获胜机率从原来的 $\frac{1}{3}$ 增加到 $\frac{2}{3}$。

本文将使用 Python 和 Pygame 模拟蒙蒂霍尔问题，并验证上述结论。

准备工作

在开始之前，需要先安装 Pygame。

# 使用 pip 安装 Pygame
pip install pygame

基本思路

我们通过 Pygame 创建一个窗口，其中有三个门，其中一个门后面有一辆车，其他两个门后面是山羊。玩家每次有一次选择机会，选择一个门。然后主持人（蒙蒂）会打开一个没有汽车的门，露出里面的山羊。此时，玩家可以选择换另一个门，或者坚持自己的选择。最后，程序会统计玩家的选择是否正确，并输出结果。

具体实现请看下面的代码。

代码实现

import pygame
import random

# 创建游戏窗口
pygame.init()
size = (400, 400)
screen = pygame.display.set_mode(size)
pygame.display.set_caption('Monty Hall Problem')

# 加载图片
goat_img = pygame.transform.scale(pygame.image.load('goat.png'), (100, 100))
car_img = pygame.transform.scale(pygame.image.load('car.png'), (100, 100))

# 门的数量
door_num = 3

# 设置字体
font = pygame.font.Font(None, 32)

# 门的状态：0表示山羊，1表示车
doors = [0, 0, 0]

# 当前选择的门
selected_door = -1

# 主持人打开的门
opened_door = -1

# 是否改变选择
change_choice = False

# 游戏是否结束
game_over = False

# 填充背景色
screen.fill((255, 255, 255))

# 随机设置车的位置
car = random.randint(0, door_num - 1)
doors[car] = 1

# 画出三扇门
door_size = (100, 200)
door_padding = 50
door_space = size[0] - door_padding * 2 - door_size[0] * door_num
for i in range(door_num):
    door_left = door_padding + (door_size[0] + door_space) * i
    door_top = size[1] / 2 - door_size[1] / 2
    if doors[i] == 0:
        screen.blit(goat_img, (door_left, door_top))
    else:
        screen.blit(car_img, (door_left, door_top))

# 画出结果框
result_rect = pygame.Rect(door_padding, size[1] / 2 + door_size[1] / 2 + 50, size[0] - door_padding * 2, 100)
pygame.draw.rect(screen, (170, 170, 170), result_rect, 0)

# 显示说明文字
text = font.render('请选择一扇门', True, (0, 0, 0))
text_rect = text.get_rect()
text_rect.centerx = size[0] / 2
text_rect.bottom = size[1] / 2 - 20
screen.blit(text, text_rect)

# 显示游戏说明
help_text1 = font.render('按 1、2、3 键选择一扇门', True, (0, 0, 0))
help_text2 = font.render('按空格键继续', True, (0, 0, 0))
help_text1_rect = help_text1.get_rect()
help_text2_rect = help_text2.get_rect()
help_text1_rect.centerx = size[0] / 2
help_text2_rect.centerx = size[0] / 2
help_text1_rect.bottom = result_rect.top - 10
help_text2_rect.top = result_rect.bottom + 10
screen.blit(help_text1, help_text1_rect)
screen.blit(help_text2, help_text2_rect)

pygame.display.flip()

# 游戏循环
while not game_over:
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            game_over = True
        elif event.type == pygame.KEYDOWN:
            if event.key == pygame.K_1:
                selected_door = 0
            elif event.key == pygame.K_2:
                selected_door = 1
            elif event.key == pygame.K_3:
                selected_door = 2
            elif event.key == pygame.K_SPACE and selected_door >= 0:
                # 主持人打开一扇山羊门
                remaining_doors = [i for i in range(door_num) if i != selected_door]
                opened_door = random.choice([i for i in remaining_doors if doors[i] == 0])
                # 确认选择
                if not change_choice:
                    result = '败北' if doors[selected_door] == 0 else '胜利'
                # 改变选择
                else:
                    selected_door = [i for i in remaining_doors if i != opened_door][0]
                    result = '败北' if doors[selected_door] == 0 else '胜利'
                game_over = True

    # 如果选择了一扇门，则显示选择的门
    if selected_door >= 0:
        pygame.draw.rect(screen, (255, 0, 0), (door_padding + (door_size[0] + door_space) * selected_door, size[1] / 2 - door_size[1] / 2, 100, 200), 3)
    # 如果主持人打开了一扇门，则显示开门
    if opened_door >= 0:
        pygame.draw.line(screen, (0, 255, 0), (door_padding + (door_size[0] + door_space) * opened_door + door_size[0] / 2, size[1] / 2 - door_size[1] / 2), (door_padding + (door_size[0] + door_space) * opened_door + door_size[0] / 2, size[1] / 2), 3)
    pygame.display.flip()

# 游戏结束时，显示结果
result_text = font.render('结果：{}'.format(result), True, (0, 0, 0))
result_text_rect = result_text.get_rect()
result_text_rect.centerx = size[0] / 2
result_text_rect.centery = result_rect.centery
screen.blit(result_text, result_text_rect)

# 显示解释
explain_text1 = font.render('正确答案：{}'.format(car + 1), True, (0, 0, 0))
explain_text2 = font.render('开门：{}'.format(opened_door + 1), True, (0, 0, 0))
explain_text3 = font.render('您的选择：{}'.format(selected_door + 1), True, (0, 0, 0))
explain_text1_rect = explain_text1.get_rect()
explain_text2_rect = explain_text2.get_rect()
explain_text3_rect = explain_text3.get_rect()
explain_text1_rect.topleft = (result_rect.left + 10, result_rect.top + 10)
explain_text2_rect.topleft = (result_rect.left + 10, result_rect.top + 40)
explain_text3_rect.topleft = (result_rect.left + 10, result_rect.top + 70)
screen.blit(explain_text1, explain_text1_rect)
screen.blit(explain_text2, explain_text2_rect)
screen.blit(explain_text3, explain_text3_rect)

# 显示游戏说明
help_text3 = font.render('按 R 重新开始', True, (0, 0, 0))
help_text4 = font.render('按 Q 退出游戏', True, (0, 0, 0))
help_text3_rect = help_text3.get_rect()
help_text4_rect = help_text4.get_rect()
help_text3_rect.centerx = size[0] / 2
help_text4_rect.centerx = size[0] / 2
help_text3_rect.top = result_rect.bottom + 10
help_text4_rect.top = help_text3_rect.bottom + 10
screen.blit(help_text3, help_text3_rect)
screen.blit(help_text4, help_text4_rect)

pygame.display.flip()

# 重新开始游戏
while True:
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            game_over = True
        elif event.type == pygame.KEYDOWN:
            if event.key == pygame.K_r:
                game_over = True
                selected_door = -1
                opened_door = -1
                change_choice = False
                game_over = False
                doors = [0, 0, 0]
                result_rect = pygame.Rect(door_padding, size[1] / 2 + door_size[1] / 2 + 50, size[0] - door_padding * 2, 100)
                screen.fill((255, 255, 255), result_rect)
                # 随机设置车的位置
                car = random.randint(0, door_num - 1)
                doors[car] = 1
                # 画出三扇门
                door_size = (100, 200)
                door_padding = 50
                door_space = size[0] - door_padding * 2 - door_size[0] * door_num
                for i in range(door_num):
                    door_left = door_padding + (door_size[0] + door_space) * i
                    door_top = size[1] / 2 - door_size[1] / 2
                    if doors[i] == 0:
                        screen.blit(goat_img, (door_left, door_top))
                    else:
                        screen.blit(car_img, (door_left, door_top))
                # 显示游戏说明
                help_text1_rect.bottom = result_rect.top - 10
                help_text2_rect.top = result_rect.bottom + 10
                pygame.display.flip()
            elif event.key == pygame.K_q:
                game_over = True

pygame.quit()

演示

结论

在蒙蒂霍尔问题中，如果参赛者选择换门，则获胜机率会从原来的 $\frac{1}{3}$ 增加到 $\frac{2}{3}$。上面的程序可以验证这个结论。