Template:Monty hall capsule: Difference between revisions

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Created page with "{{quote| 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..."
 
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{{quote|
{{quote|
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 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.


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


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


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


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


Bayesian inference shows that intuition to be ''wrong''.  
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. 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.
So, while your odds have ''improved'', the odds remain equal for each unopened door. So, it still doesn’t matter which you choose: ''Right''?


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.
''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.  


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.  
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.


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.  
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.  


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.
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.

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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