FILE No.79Unsolved
Behind one of three doors (1, 2, 3) is treasure, and the other two are empty. The host knows where the treasure is.
The contestant picks door 1. The host always opens an empty door among the other two, and always offers a switch. This time he opens door 3, which is empty, and asks, "Would you like to switch to door 2?"
If you switch to door 2, what is the probability that you find the treasure?
If you get stuck, open the hints one at a time, in order.
What is the chance that your first pick is right?
Could the host ever open the door with the treasure?
Where does the remaining probability go?
If you switch to door 2, what is the probability that you find the treasure?
Choose the answer closest to the truth.
2/3
The answer is: 2/3.
The chance that your original pick, door 1, has the treasure is 1/3. So the chance that the treasure is behind door 2 or door 3 is 2/3. Because the host knows where the treasure is and always opens an empty door, all of that 2/3 moves onto the one unopened door, door 2. Switching wins with probability 2/3.
It is tempting to say "two doors left, so 1/2." But the host knows where the treasure is and opens an empty door on purpose, so the probabilities are not equal.
Intuition says "two doors left, so fifty-fifty."
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