Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a sports car; behind all of the others is bicycles. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 2, revealing a bicycle. He then says to you, “Do you want to pick door No. 3?”

Is it to your advantage to switch your choice?

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In short, it depends on some assumptions regarding the hosts behaviour (see wikipedia article). But if we follow the standard assumptions then surprisingly it is of benefit to switch as when you switch under these assumptions you have a 2/3 chance of winning the sports car vs 1/3 if you stick!

I remember first wrapping my head around this one. The trick to me and many people finally “getting it” is to not think of the host opening a single other door, rather consider that he opens ALL doors EXCEPT one. Either the prize is behind the door you first chose, or its behind the one remaining door. With only 3 doors it seems like the same thing, but isn’t.

Consider there are 100 doors. You have a measly 1/100 in picking the right door in one guess. Now the host opens all the other doors except one. Either you got lucky and chose the right door first, or the one of 99 left that he didn’t open contains the prize. Obviously the latter is very much more likely.

Yes, pick door number 3. I saw this on the TV NUMB3RS one time.

It doesn’t matter

Andy, you are today’s winner!

Yes, by changing your answer your chances of winning actually goes up from 1/3 to 2/3. This becomes obvious when expanding the example. Suppose there was 100 doors rather than 3. You pick one and the host shows you that the car is not behind 98 of the doors then asks you to switch to the remaining door or keep the door you picked. Of course you would switch your door because chances are you didn’t pick the correct door initially.laskhdbsmclgljnb LKuglKb.kDUHX,djkbzxiugclkzjldgBkjDBflbdf.bZLYUfgLYGFkljxbkjgvioydGfxlZKJvflsadgfsidbvliuxykdsnf.kdjshgfllaksdglaskjflaslaksdjalsdlasdhlajksdfkdashfgelgyalwiufsdkjhdlsbliyeaglfksjdgflieygslkglfkesabvugifsfalkjfg

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