Four doors, one peek

Question

There are 4 doors and one prize. You choose a door and keep it closed. The host, who knows where the prize is, opens exactly one empty door from the other three. That leaves your door plus two other closed doors. You decide to switch, picking one of those two remaining doors at random.

What is your probability of winning the prize?

Accepted Answer

Split on whether your first pick was correct. - Your door hides the prize with probability 	frac{1}{4}. Then both other closed doors are empty, so switching to either loses. - Your door is empty with probability 	frac{3}{4}. Then the prize is among the other three doors. The host opens one empty door from them, so the prize is still behind one of the two closed doors you may switch to. Picking one of those two at random wins with probability 	frac{1}{2}. Combine: P(	ext{win by switching}) = \frac{1}{4}\cdot 0 + \frac{3}{4}\cdot\frac{1}{2} = \frac{3}{8}. Since staying wins only 	frac{1}{4…

Four doors, one peek

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