Problem #6 MEDIUM
The Monty Hall Dilemma
Meta Logic Math
Problem Statement
You are a contestant on a game show. The host presents you with three closed doors. Behind one door is a brand new sports car; behind the other two are goats.
You choose a door—say, Door 1. The host, who knows what is behind each door, opens one of the other doors—say, Door 3—revealing a goat. He then asks you: "Would you like to switch your choice to Door 2?"
Is it to your advantage to switch your choice, or does it not matter?
Answer & Quick Explanation
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Yes, you should always switch. Switching gives you a 2/3 chance of winning the car, while sticking with your initial choice keeps your odds at 1/3.
Detailed Editorial Solution
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This is the famous Monty Hall problem. At first glance, it seems that with two closed doors remaining, the probability must be 50-50. However, this is incorrect because the host's action is not random.
Let's break down the probabilities:
1. When you first choose Door 1, there is a 1/3 chance you selected the car, and a 2/3 chance the car is behind one of the other two doors (Door 2 or Door 3).
2. The host MUST open a door with a goat. By doing so, he consolidates the entire 2/3 probability of the "other doors" group into the single remaining unopened door (Door 2).
3. If you stick with Door 1, you win only if you were right initially (1/3 chance).
4. If you switch to Door 2, you win if you were wrong initially (2/3 chance).
Therefore, switching doubles your chances of winning from 1/3 to 2/3.