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The Infamous Monty Hall Problem: Answer
Hold on to your hats...
you *double* your chances by switching
This is, at first look, way counter-intuitive, so here's an attempt at an
explanation:
Take a look at this matrix of possibilities:
| |
Door A |
Door B |
Door C |
| Case 1: |
bogus |
bogus |
Good |
| Case 2: |
bogus |
Good |
bogus |
| Case 3: |
Good |
bogus |
bogus |
Let's assume you choose door A --
you have a 1/3 chance of a Good prize.
But (this is key) Monty knows what is behind each door,
and shows a bogus one.
In cases 1 and 2, he eliminates doors B and C respectively
(which happen to be the only remaining bogus door)
so a Good door is left: SWITCH!
Only in case 3 (you lucked out in your original 1 in 3 chances)
does switching hurt you.
So, your probability goes up from 1/3 to 2/3
if you switch after being shown a bogus door.
Caveat: of course, this only works if Monty is guaranteed
to show you a bogus door every time after you choose a door,
something that was not assured in the original game show.
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