Answer to the Monty Hall Problem

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
		      ~~~~
	case	A	B	C
	~~~~
	1	bad	bad	good
	2	bad	good	bad
	3	good	bad	bad
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 bad one.

In cases 1 and 2, he eliminates doors B and C respectively (which happen to be the only remaining bad 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 bad door.

Caveat: of course, this only works if Monty is guaranteed to show you a bad door every time after you choose a door, something that was not assured in the original game show.

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