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BGonline.org Forums
My best probability puzzle
Posted By: Max Stockslager In Response To: My best probability puzzle (Timothy Chow)
Date: Wednesday, 21 August 2024, at 12:56 p.m.
After I got over my initial reaction (“how could either of them possibly have an advantage”), I had the same intuition as Bob Koca that Andrew must win more often because Barbara wastes more guesses by picking boxes that Andrew has already found are empty. Except I couldn’t figure out why this didn’t also give Andrew an advantage in the n=1 prize case. What I didn’t appreciate until walking through a 2x5 toy example was that Andrew wastes just as many guesses as Barbara, but his wasted guesses happen later in the game. In the n=1 case the prize is equally likely to be in the one of the four boxes that Andrew opens first or one of the four boxes that Barbara opens first (the other two boxes tie) so neither player has an advantage. However, if there are 2 or more prizes, each player will still on average have the same number of prizes in the set of boxes they will open first (1 prize each), but because Andrew gets a head start on opening the boxes he will reach first, the game will usually end before Barbara finishes opening many of boxes she will reach first (since these come later) which gives Andrew an advantage. This last argument could probably be strengthened but I think it is at least intuitive to me now why this only applies when there are multiple prizes.
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