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 carries an equal probability: ⅓, After the reveal, the remaining doors still do: ½.

So, while your odds have improved, the odds remain equal for each unopened door. So, it still doesn’t matter which you choose: Right?

Wrong. The best odds are if you switch: there remains a ⅓ chance the car is behind the first door you picked; there is now a ⅔ chance the Ferrari is behind the other door. Staying put is to commit to a choice you made then the odds were worse.

We know this thanks to Bayesian inference. There are two categories of door; ones you chose, and ones you didn’t. There’s only one door in the “chosen” category and two doors in the “unchosen” category. At the start you knew each was equally likely to hold the car. This was the “prior probability”. There was a ⅓ chance per door or, if we categorise the doors, a ⅓ chance it was behind a chosen door and a ⅔ chance it was behind an unchosen door.

Then you got some updated information, but only about the “unchosen door” category: One of those doors definitely doesn’t hold the car. You have no new information about the “chosen door” category, however.

You can update your prior probability estimates about the unchosen doors. One now has a zero chance of holding the car. Therefore, it follows the other door has a ⅔ chance. All the odds of the unchosen category now sit behind its single unopened 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.