When the host forgets

Question

Three doors hide one car and two goats. You pick door 1. A new assistant who does not know where the car is opens one of the other two doors purely at random, and it happens to reveal a goat. The assistant opened door 3.

What is your probability of winning the car if you now switch to door 2?

Accepted Answer

This looks like the usual game, but the assistant acted blindly, so his lucky reveal carries different information than a knowing host's would. Work through where the car can be, each equally likely at the start. - Car behind door 1 (your pick): the assistant opens door 2 or door 3 at random. He opens door 3 and it is empty. This path has probability 	frac{1}{3}\cdot	frac{1}{2}=	frac16. - Car behind door 2: the assistant opens door 2 or door 3 at random. Opening door 3 shows a goat (car is at 2). Probability 	frac{1}{3}\cdot	frac{1}{2}=	frac16. - Car behind door 3: if the assistant open…

When the host forgets

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