Monty Hall

Switch your door — P doubles. The reveal is informative, not symmetric.

The idea

The Monty Hall problem is the standard example of how probabilities should be revised when someone who knows more than you reveals something. A prize sits behind one of several doors. You pick one; the host, who knows where the prize is, opens losing doors from among the others until just one of them remains closed, and offers to let you switch to it.

How far a revelation should move your beliefs depends on how likely that revelation was under each possibility you are weighing. A revelation that was bound to happen, whichever possibility is true, distinguishes nothing and leaves the probabilities where they were.

The tempting error is to count the options left standing and split the probability evenly among them. That is correct only when the revelation was equally likely under each possibility, and it fails here because the host's choice is constrained by his knowledge.

So fix each probability before anything is revealed, while the doors are still symmetric, and then ask what the host was forced to do. The probability of a door the host eliminated does not spread out evenly; it moves onto whatever his constraint left closed.

Ways to work on it

• Walkthrough. Why switching wins with probability 2/3, step by step.
• Practice. Compute the switching probability with a few more doors.
• Hardest. Compute the switching probability when the number of doors is large.

Not sure where to start? Take the ten-question placement test.