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Monty Hall Problem

The Three Doors: Switch or Stay?

📚 Introduction

🚗 Three Doors Problem

A classic probability puzzle from a TV game show. The rules are simple:

  1. Behind three doors: one car (prize), two goats
  2. You pick a door
  3. Host (who knows) opens another door with a goat
  4. You can: stick with your choice, or switch to the other door

❓ The Question

Switch or stay? Which strategy has better odds?

🤔 Intuition vs Math

大多数人直觉认为换不换都是50%的概率。但数学告诉我们：换门赢的概率是2/3，不换只有1/3！ 这个反直觉的结论曾引发激烈争论，连数学家都曾犯错。

💡 Why is switching better?

关键在于：你最初选中汽车的概率只有1/3，选中山羊的概率是2/3。 当主持人揭示一扇山羊门后，如果你最初选的是山羊（2/3概率），换门必定赢； 只有当你最初就选中汽车（1/3概率）时，换门才会输。

🎮 Try It Yourself

Pick a door

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📊 Batch Simulation

🎲

Click to run batch simulation

Verify the switching advantage with data