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.

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?

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?

Can you generalize this for an NxN checkerboard?
Source: Original.
Solution
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