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BGonline.org Forums
My best probability puzzle
Posted By: Timothy Chow
Date: Friday, 16 August 2024, at 12:32 p.m.
The following probability puzzle is original with me. You can find it (phrased slightly differently) in the recently published book, The Mathematical Playground, by Alissa Crans and Glen Whitney.
Andrew and Barbara are playing a game. Fifteen boxes are arranged in a 3-by-5 grid, labeled with the letters A through O, as shown below.
A B C D E
F G H I J
K L M N O
The organizers of the game have chosen two boxes randomly and placed a prize in each of the two chosen boxes. (It is the same prize in both boxes.) Andrew and Barbara each secretly submit to the organizers a search order, meaning the order in which they wish to examine the boxes. Andrew decides to search the boxes row by row, from left to right, top to bottom. His search order is therefore ABCDEFGHIJKLMNO. Barbara, however, decides to search the boxes column by column, from top to bottom, left to right. So her search order is AFKBGLCHMDINEJO.
After receiving the two search orders, the organizers now step through the two search orders in parallel. The first player to hit one of the prizes is declared the winner (ties are possible). For example, suppose that K and E happen to be the boxes with prizes in them. Then Barbara will hit the prize in box K on step 3, whereas Andrew won't hit a prize until step 5, when he reaches box E, so Barbara wins in this case. For another example, suppose that the prizes are in boxes H and N. Then both Andrew and Barbara will hit the prize in box H on step 8, and the result will be a tie. For a third example, suppose that the prizes are in boxes E and G. Then Andrew will reach box E on step 5 and Barbara will reach box G on step 5, and again it will be a tie.
QUESTION: Is Andrew more likely to win this game, or is Barbara? Or are they equally likely to win?
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