蒙提霍尔问题
Monty Hall Problem
题目详情
你参加一个游戏：三扇门后面一扇是汽车，另外两扇是羊。
你先选一扇门（比如 1 号门）。主持人知道每扇门后是什么，会打开另一扇有羊的门（比如打开 3 号门）。
此时你可以选择是否改选剩下那扇没开的门（2 号门）。问：如果你选择换门，赢得汽车的概率是多少？
提示：主持人总是打开一扇有羊的门。
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. Now, do you want to pick door No. 2? What is the probability to win the car if you switch?
Hint
The host always opens a door with a goat.
解析
换门赢车概率为 。
初始选中汽车的概率是 ，选中羊的概率是 。
主持人必定打开一扇羊门，因此当你一开始选中羊（概率 ）时，剩下未开的门一定是汽车；换门就赢。故换门胜率为 。
Original Explanation
2/3
Solution
The probability that your initial choice did not have a car is indeed .
Initial Misstep: After one door is opened, there are exactly two doors left, and one of them has a car. So the probability that the car is behind either door is . This is incorrect because the host knows which door has a car and which door has a goat. The host always opens a door with a goat.
Correct Solution: The following table shows which Door the host might open. Assume that the car is behind Gate #1, and we randomly choose one door.
| Initial choice: | Door #1 | Door #2 | Door #3 |
|---|---|---|---|
| Reality | Car | Goat | Goat |
| Host opens: | #2 or #3 | #3 | #2 |
| Remaining: | #3 or #2 | #1 | #1 |
| Good to switch: | No | Yes | Yes |
We see that at the end, the remaining unopened door is Door #1 if we start with Door #2 or Door #3. This means that in 2 out of 3 cases, we started with the incorrect Door (#2 or #3) and we got the option to switch with the correct door at the end (#1).
So you should switch to the other door, and win the car with a probability of
Generalization
The probability of being initially wrong is the same as the probability of being correct after switching. We can generalize this to doors. The probability of winning the game by switching after the host has opened doors is
Script
Not convinced? Simulate using this Colab Script