Game Show Puzzle

michaellevenson

Member: Rank 8
There are three doors, with a prize behind just one of them. The contestant picks a door but does not open it yet. She knows that the game host, Monty Hall, knows where the prize is and, when he opens another door to reveal nothing behind it, that he has used his knowledge to do so. Monty Hall then offers the opportunity of changing doors and swap for the one remaining closed door. Should she accept?

Monty Hall, US Game Show Host.
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michaellevenson

Member: Rank 8
This puzzle simply illustrates the popular confusion about probability. Its not much of a puzzle tbh, however some academic mathematicians vehemently refused to accept swapping was advantageous, including great number theorist Paul Erdős. He was eventually convinced after a computer run simulated the puzzle 300 times, and 200 times swapping proved the winning outcome.
Either the other unopened door conceals the prize or the first one picked does. Doesn't that make it equally likely that either of those two doors has the prize behind it?
It does not. When the contestant first picks a door the chance that it has the prize is ⅓. She knows that Monty Hall will be able to open a door concealing no prize, since at least one of the other doors must be a loser.
Hence she learns nothing new which is relevant to the probability that she has already chosen the winning door: that remains at ⅓. Since if she swaps she will not choose the door Monty has just revealed to be a loser, the opportunity to swap is equivalent to the opportunity of opening both the other doors, which clearly doubles her chances of winning.
It would be different if another contestant picked a door which turned out to be a loser. That would raise the chance that her door was the winner from ⅓ to ½. By contrast with Monty Hall whose chance of opening a losing door was 1, since he knew where the prize was, the other contestant's chance is only ⅔. She would still have opened it had it contained the prize. The unopened doors are now equally likely.
How can this be if in both cases you learn that one of the other doors is a loser?
Unless the contestant has picked the winning door originally, Monty Hall has used his knowledge to isolate the winning door for her by opening the other empty one. But the other contestant who opens a door which happens to be a loser has no knowledge of the content of the unpicked door to guide her in her choice. So it remains likely to be the winner as the one originally picked by the first contestant.
The contrast is easier to see if we imagine 100 doors, just one containing a prize.
The first contestant picks a door then Monty Hall using his knowledge of where the prize is opens 98 losing doors. The contestant has a 1% chance of having picked the prize door. If she hasn't, Monty has isolated the prize by opening 98 losing doors. By accepting a swap her chance of winning increases to 99%. If on the other hand, the other contestant opens 98 doors all losers, the chance the first contestant has the winning door increases until it's 50%. The other contestant cannot be isolating the winning door as she doesn't know where the prize is.
What if the contestant does not know whether Monty Hall knows the location of the prize, and if so whether he'll make use of the knowledge?
Unless she is certain that he doesn't know or make use of it, it still makes sense to swap since there is a chance he has isolated the winning door for her. Swapping will not double her chance of winning but it will raise it to between ½ to ⅔, which will therefore be greater than the original probability. From the book, A to Z of Paradoxes by Michael Clark.
 
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