Template:Monty hall capsule: Difference between revisions

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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.
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.<ref>''Why goats? — Ed''</ref> You may choose one door.


Knowing you have a ⅓ chance, you choose a door at random.  
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.}}
Now the host theatrically opens one of the doors you ''didn’t'' choose, revealing a goat.  


Do you stick with your original choice, switch, or does it not make a difference?
Two closed doors remain. She offers you the chance to reconsider your choice.


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


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


Staying put is to commit to a choice you made then the odds were worse.  
So, while your odds have ''improved'', the odds remain equal for each unopened door. So, it still doesn’t matter which you choose: ''Right''?


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.
''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.  


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.
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.


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.  
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.  


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


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

Latest revision as of 18:48, 6 November 2024

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.[1] 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.

  1. ↑ Why goats? — Ed
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