The Monty Hall problem
Asked at Jane Street, SIG, IMC
You're on a game show with three doors. Behind one is a car; behind the other two are goats. You pick a door, say Door 1. The host, who knows what's behind every door, opens a different door (say Door 3) revealing a goat. He always does this: he always opens a losing door you didn't pick. He then offers you the chance to switch to the remaining unopened door (Door 2).
Should you switch, stay, or does it not matter? What is your probability of winning if you switch?
Show a hint
Your original pick was made when all three doors were equally likely. Does the host opening a goat door change the fact that your first door had a 1/3 chance?