Template:Monty hall capsule

You are a game-show contestant. The host shows you three doors and tells you: “Behind one of those doors is a Ferrari. Behind the other two are goats. You may choose one door.

Knowing you have a ⅓ chance, you choose a door at random.

Now the host theatrically opens one of the doors you didn’t choose, revealing a goat.

Two closed doors remain. She offers you the chance to reconsider your choice.

Do you stick with your original choice, switch, or does it not make a difference?

Intuition suggests it makes no 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. The odds are now ½ for each remaining door. Right?

Wrong. Bayesian reasoning shows our intuition is wrong. The best odds are if we switch.

Staying put is to commit to a choice you made then the odds were worse.

There are two categories of door; the one you chose, and ones you didn’t. Because there is only one door in the “chosen” category and two doors in the “not chosen” category, and each was equally likely to hold the car, we know the odds between the categories were: chosen: ⅓; not chosen: ⅔.

Now we get some updated inormation: Monty opens one of the “not chosen” doors. This is gives you no more information about the chosen door but it does give you information, about the unchosen doors: we know one of them is empty. It definitely has a zero chance of containing the car. Therefore, all the odds of the unchosen category, sit behind the unopened door.

The “chosen” door still has the ⅓ chance it started with. The “not chosen” category remains at ⅔ but it is all vested in the one remaining unchosen door.

Therefore you have a better chance of winning the car (though not a certainty — one time in three you’ll lose) if you switch.

A person who now arrives, with two doors remaining, who is given the choice without your prior knowledge, would calculate the probabilities at 50:50. But she is ignorant of your original choice and the decision the host made, based on your original choice (remember the door opened by the host depended on your choice: the rule was “open a door that the contestant did not choose and that does not conceal the Ferrari”). Without that information, from the newcomers perspective, the odds really are 50:50.

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.