Ball Drop
Place pins at the corners of all the squares on an 4x4 checkerboard. Now cut the
board in half across a main diagonal. Then
lift the board to hold one corner at the top and the long diagonal across the
bottom. Drop a ball at the top pin. When it bounces, it has a 50%
chance of heading left or right to the next lower pin, where it again has a 50%
chance of a left or right move, and so on until it exits the
board.
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There are 6 different locations the ball could exit the board (the four diagonal
squares, and outside the board on either side.) What are the
probabilities of exiting from each of them?
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Now replace the other half of the board. In how many different
locations could the ball exit the board, and what are the probabilities of
exiting from each of them?
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Can you generalize this for an NxN checkerboard?
Source: Original.
Solution
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