Monty Hall

Question

Three doors: one hides a prize, two hide nothing. You pick a door. The host, who knows where the prize is and always opens an empty door you didn't pick: opens one. Should you switch to the remaining door, and what is your probability of winning if you do?

Accepted Answer

Switch. Winning by switching happens exactly when your initial pick was wrong: P(textwin by switching) = P(textinitial pick wrong) = frac23. The cleanest framing: your door was chosen before any information flowed, and the host's behavior is deterministic given the prize location, he always can and must open an empty other door, so his reveal carries no information about your door. Your door stays at frac13; the eliminated door's probability collapses onto the one remaining door, which now carries frac23. The assumption that matters If the host had opened a random unpicked door that merely happened to be empty, the answer becomes frac12, conditioning on a lucky reveal is different from conditioning on a forced one. Interviewers push on exactly this distinction ("what if the host sometimes doesn't offer a switch?"), because in markets the analogous question is why someone is showing you a price.

Monty Hall

Problem

Your answer

Hints

More Probability to train on