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In a game, a prize is hidden behind one of five doors, chosen at random…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
In a game, a prize is hidden behind one of five doors, chosen at random. A player picks one door. The host knows where the prize is. He opens two of the other four doors, always choosing doors without the prize, and he chooses at random among the doors that he is allowed to open. Two other doors stay closed, besides the player's door. The player may now keep the first door or switch to one of the two other closed doors, chosen at random. Find the probability that the player wins by keeping the first door. Find the probability that the player wins by switching to one of the two closed doors, chosen at random. Find the probability that the prize is behind one of the two other closed doors taken together.
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The worked solution
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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. Keeping. The first pick is right with probability . The host never opens the prize door and never opens the player's door, so he gives no information about the player's door.
Step 2. The other doors together. The prize is among the other four doors with probability . The host removes two empty doors and always leaves the prize door closed if the prize is among the four. So the two doors that stay closed hold the prize with probability together.
Step 3. Switching at random. By symmetry the two closed doors are equally likely. Each holds the prize with probability .
Answer: , and .
Check. The three possibilities add up: the first door , plus the two closed doors , give . A player who switches at random doubles the chance of winning, from to .
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