Template:Monty hall capsule

You are a game-show contestant. The host asks you to choose a prize from behind one of three doors. She tells you: behind one door there is a Ferrari. Behind the other two are goats. [Why goats? — Ed] Choose your door.

You choose a door.

Before opening your door, the host theatrically opens one of the other two doors, and reveals a goat. She offers you the chance to reconsider.

Would you reconsider?

Intuition suggests it should not make a difference. At the beginning, each door carried an equal probability, ⅓, and after the reveal, the remaining doors still do: ½. So, while your odds have improved, it still doesn’t matter. Whether you stick or twist, you should be indifferent.

Bayesian inference shows that intuition to be wrong.

Staying put is to commit to a choice you made then the odds were worse. You have no more information about the choice you made: you already knew it may or may not contain the car. You do, however, know something new about one of the doors you didn’t choose. The odds as between the other two doors change, from ⅓ each to nil for the open door — it definitely doesnʼt hold the car — and ⅔ for the other closed one, which still might.

The remaining probabilities are therefore ⅓ for your original choice and ⅔ for the other door. You have a better chance of winning the car if you switch.

It is true: a person who now arrives and is given the choice without your knowledge, would calculate the probability at 50:50. The calculation would be wrong because an important assumption in calculating probabilities — that the car and goat were normally distributed between two doors — does not hold. A third door has been unrandomly eliminated.

So you should switch doors. This proposal outrages some people, at first. Especially when explained to them at a pub, it outrages them later. But it is true.

It is easier to see if instead there are one thousand doors, not three, and after your first pick the host opens 998 of the other doors.