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. [Why goats? — Ed] you may choose one door.

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

Before revealing what is behind your door, the host stops, and theatrically opens one of the other doors, to reveal a goat. There are two closed doors left. 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; ones 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 at the beginning all doors were equally likely to hide the car, we know the odds between the categories were: chosen: ⅓; not chosen: ⅔.

Then Monty opens one of the “not chosen” doors. You have no further information about your chosen category but you do, however, about the unchosen category: one of the two is empty, and definitely has a zero chance of containing the car. Therefore, of the unchosen category, ’'all the chance sits behind the unopened door.

The6⁷u probabilities between the chosen and not chosen categories remain ⅓ and ⅔: it is just that so if the ⅔ chance is vested hlin the one remaining unchosen door. You have a better chance of winning the car (though not a certainty) if you switch.

It is true: a person who now arrives and 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.