Definition
The Monty Hall problem is a probability puzzle inspired by the American game show Lets Make a Deal, hosted by Monty Hall. A contestant picks one of three doors; behind one is a car, behind the others goats. The host, who knows what is behind every door, opens one of the unchosen doors to reveal a goat, then offers the contestant the chance to switch. Should they switch?
The counter-intuitive answer is yes: switching wins with probability 2/3, while sticking wins with probability 1/3. The result, popularised by Marilyn vos Savant in 1990, generated enormous controversy — even from credentialed mathematicians who insisted she was wrong.
Why it matters
How it works
The cleanest way to see why switching wins 2/3 of the time is to notice that your initial pick was correct 1/3 of the time (in which case switching loses) and incorrect 2/3 of the time (in which case the host's reveal points exactly at the car, and switching wins). The host's action removes a wrong door from the pool of unchosen doors, transferring its probability mass to the remaining one.
A more formal argument uses Bayes' theorem with explicit sample-space enumeration of doors-and-host-choices. Both methods give 2/3 for switching. The puzzle's lasting power is that the intuition — 'now it is 50-50 between two doors' — is exactly the type of reasoning that conditional probability is built to correct.