079概率中等数值题medium

Four-Door Monty Hall

题目

There are four doors: one hides a car, the other three hide goats. You pick Door 1. The host, who knows where the car is, opens one of the remaining doors to reveal a goat (choosing uniformly at random among the goat doors he can open). He opens Door 4. You are now given two options: (a) stick with Door 1, or (b) switch to one of the two unopened doors (Door 2 or Door 3), chosen uniformly at random. What is the probability of winning the car under each option? Is there a third strategy that does even better?

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你的答案

Probability of winning when sticking with Door 1

Probability of winning when switching randomly to Door 2 or Door 3

Probability of winning with the best strategy (switching to a specific door)

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Generalize: with $n$ doors, you pick one, and the host reveals $k$ goat doors ( $1 \leq k \leq n-2$ ). If you switch uniformly among the remaining $n-1-k$ doors, what is your winning probability?

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