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You are on a gameshow. You are presented with three doors. Behind two of the doors are goats, behind the third is a sportscar.
You choose a door. The presenter then opens one of the other doors, revealing a goat. You now have the option of choosing to open the original door you selected, or changing your mind and opening the other remaining door. Whatever is behind the door you open, you get to keep. Which is the better choice?
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Is there anyone in this day and age who wouldn't immediately google for "Monty Hall problem"? Such problems were cool once, like, 30 years ago...
Well, this is a bumped thread from 2005. Yeah, I thought the Monty Hall problem was a settled question by now. I like one of Cecil's counterarguments in Bloodshedder's Straight Dope link though. In the actual game show this brainteaser is based on, did Monty Hall always open a door or did he sometimes leave them closed just to mess with people?
I don't get it. Let's say I have to guess a letter of the alphabet. I pick A. The chance that I'm correct is 1/26. Now it is revealed that the right letter is either A or B. Are you going to say that my chances are still only 1/26? That makes no sense I think.
The key point here that screws up most people is that you can't just consider the final state, when there are just two letters left, as an independent problem. To correctly calculate the odds of winning you need to start with the initial conditions, when every letter has a 1 in 26 chance of being right, and update those individual probabilities with new info as it's revealed to you.
For the sake of illustration let's pretend that the person giving you this challenge - I'll call them the "dealer" - doesn't know which letter is correct either. Let's say that all 26 letters are written on the backs of face-down cards, one of which has a coupon for 25% off an Arby's roast beef sandwich printed on the front. You take the "A" card, and then the dealer just starts picking up other cards and turning them over at random. If "Q" turns out to be blank, then you can use that info to update the probabilities: every card now has a 1/25 chance of being the winner. If "M" is also blank, the odds become 1/24. If, somehow, only "A" and "B" are left at the end and the coupon still hasn't been found, both letters have a 50/50 chance of hiding the prize. Of course, this in itself is a pretty unlikely outcome. There's only a 1/13 chance that it would come down to two final cards like this - usually the coupon will turn up sooner. Overall, there's only a 1/26 that the game will come to a proper two-card showdown AND your choice will prove to be right.
Now, in the Monty Hall version of this problem, the dealer isn't just flipping cards over randomly. He knows exactly which letter is the winner, and this is the key fact to keep in mind when you're updating the probabilities. Right at the beginning, when you pick "A," it has a 1/26 chance of being the winner, and there's a 25/26 chance that the prize is somewhere in the rest of the alphabet. Then, the dealer flips Z upwards, and it's blank. With 24 more cards in the dealer's stash, what are the odds that one of them is the coupon? 25/26. Why? Because you know that the dealer is consciously avoiding the winner. There was a 1/26 chance that "A" would win when you picked it, and now the dealer is just clearing garbage from his side of the table, with no new random events happening and no modification of the odds. There's a 0/26 chance that the game will end early with an accidental prize reveal, and a 26/26 chance that it will end with a choice between two cards. At that point, you can either take the 'A' card with a 1/26 chance of victory, or the dealer's final card, which has a 25/26 chance of giving you a deal on a sandwich lovingly stuffed with saucy meat.
These proofs can persuade me mathematically but not in spirit.
I like the way you put that. That was my definitely my reaction the first time I heard about this problem. Let the math soak into your subconscious and maybe it'll make more intuitive sense the next time you stumble across a random mention of "Monty Hall" a few years from now.