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53 lines
2.5 KiB
Markdown
53 lines
2.5 KiB
Markdown
---
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lastmod: "2021-09-12T22:47:44.0000000+01:00"
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author: patrick
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categories:
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- mathematical_summary
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comments: true
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date: "2016-04-08T00:00:00Z"
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aliases:
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- /another-monty-hall-explanation/
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title: Another Monty Hall explanation
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---
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Recall the [Monty Hall problem]: the host, Monty Hall, shows you three doors, named A, B and C.
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You are assured that behind one of the doors is a car, and behind the two others there is a goat each.
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You want the car.
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You pick a door, and Monty Hall opens one of the two doors you didn't pick that he knows contains a goat.
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He offers you the chance to switch guesses from the door you first picked to the one remaining door.
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Should you switch or stick?
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I'll slightly reframe the problem: let's pretend you are playing cooperatively with Monty Hall, where he
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knows the layouts and he is trying to open two goat-doors, and you're trying for the car; you're not allowed to communicate.
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The game is (noting the distinction between "picking" a door - i.e. announcing your intention to open it - and opening it):
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* You pick a door;
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* Monty Hall opens a door you didn't pick;
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* You open a door Monty Hall didn't just pick;
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* Monty Hall opens the remaining door.
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(The problem is the same: in standard Monty Hall, you win if and only if you open the car door and Monty Hall opens two goat doors.
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Let's say Monty Hall really likes goats, and not inquire further.)
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You pick a door, B say. Monty Hall now opens a goat-door, C say,
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because he knows the layouts and can pick one with a goat behind.
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At this point, you know Monty Hall *decided not to open* door A.
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Why would he not have chosen door A?
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It's either because he chose randomly between his available goaty options A and C,
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or because he knew A had a car behind so he was choosing the only goat door available to him.
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(Remember, Monty Hall wants to find goats.)
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If he chose randomly, you're better off sticking, because that means you have the car.
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But if he *actively refused* door A (which can only happen because it had a car behind), that means you need to switch to door A.
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He chose randomly with probability 1/3 (because he chose randomly if, and only if, you originally picked the car).
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He actively refused door A with probability 2/3, therefore.
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So with 2/3 probability, you're in the case that means you guarantee yourself a car if you switch.
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With 1/3 probability, you're in the case that means you guarantee yourself a car if you stick.
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So you should switch.
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[Monty Hall problem]: {{< ref "2013-12-22-three-explanations-of-the-monty-hall-problem" >}}
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