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The Monty Hall problem

You are in a game show. You have to choose between three buttons, A, B and C. Pressing one of them will give you £200,000, and pressing either of the other two will give you a free mousemat. You choose a button at random (button A). The gameshow host doesn't tell you if you have won or not, but he does tell you that button B was one of the wrong buttons. He also gives you a chance to change your mind. Should you stick with button A, or switch to button C? Why? Or does it make no difference?

Can any one suggest what should be the possible answer for this. Does it really make any difference If I change my mind? So what are the possibilities here?

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migrated from Jun 21 '12 at 21:29

This question came from our site for theoretical computer scientists and researchers in related fields.

marked as duplicate by Arturo Magidin, Marvis, mixedmath, David Mitra, Byron Schmuland Jun 21 '12 at 21:57

This question was marked as an exact duplicate of an existing question.

See Although I'm torn between a mousemat and a donkey. – talmid Jun 21 '12 at 21:36
Look up the "Monty Hall problem". Your description is not specific enough to give an answer: does the host always offer you a choice to change your mind? Does the host know where the money is and always reveal an unchosen button as incorrect? Etc. This site alone offers several questions featuring this problem: here, here, here. See also Wikipedia. – Arturo Magidin Jun 21 '12 at 21:36
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This is an very well-known problem.

Your probability of success is maximised by changing. You can read about it all day here: ,

but it is instructive to think about it yourself if you are actually studying probability. Writing out rigorously why switcihing is better can be tricky.

The one intuitive explanation that really stuck with me is the thought experiment in which you imagine there are, say 100 buttons. You pick one, say button No. 1. The gameshow host reveals to you that buttons 2 through 56 and buttons 58 through 100 are wrong buttons. Now do you want to switch to door 57 which for some reason the host has not mentioned? Or do you really think you would do just as well sticking with button No. 1? It seems clear to me now that in sticking with door No. 1 you have a 1/100 chance of being correct.

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