The Full Monty Hall

One of my students, who endured my application of Bayes’ Rule to the Monty Hall problem, reminded me that Marilyn vos Savant addressed the problem once again in last week’s column:

You are on a game show with three doors. A car is behind one; goats are behind the others. You pick door No. 1. Suddenly, a worried look flashes across the host’s usually smiling face. He forgot which door hides the car! So he says a little prayer and opens No. 3. Much to his relief, a goat is revealed. He asks, “Do you want door No. 2?” Is it to your advantage to switch?
—W.R. Neuman, Ann Arbor, Mich

Nope. If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch.

Correct answer. However, you should switch regardless. Why? If Monty is even slightly more likely to pick a door with a goat behind it (maybe he didn’t completely forget, he has a hunch the car is behind door #2...), then your odds are better if you switch.

For the same reason, you should switch if you don’t know which policy Monty is following. If he chooses randomly (50-50), switching doesn’t hurt you; and if he favors opening doors with goats, switching helps you. The only reason not to switch is if you think your host might be Malicious Monty, who deliberately favors opening the door with the car.