2. You are a guest on a television game show. There are three doors, behind one

Question

2. You are a guest on a television game show. There are three doors, behind one of which lies a valuable prize, which you will win if you select the correct door. Assume you decide to pick door 1. Then, the host opens the door 3 and reveals that there is no prize behind it. He then offers you chance to switch to Door 2? Should you switch? a) Assume that the host never opens the door behind which the prize lies, and never opens the door which you have selected (door 1). (Hint: think about the possible cases and find the conditional probabilities accordingly.) b) Assume that the host is able to select the door with the prize behind it. How does the solution changes?

Explanation / Answer

You should switch doors. There are 3 possibilities for the first door you picked:

Each of these cases are equally likely. So if you switch, there is a 2/3 chance that you will win (because there is a 2/3 chance that you are in one of the first two cases listed above), and a 1/3 chance you'll lose. So switching is a good idea.

If the host guesses the answer correclty, then it makes no difference what the game show host does, the other door is always the wrong door. So in that case, by keeping your choice, the probability that you win is 1/3 x 1 = 1/3.

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