The Galton board is a triangular array of pegs. The rows are numbered 0, 1, ..., N from the top row down, such that row n has n + 1 pegs. Suppose that a ball is dropped from above the top peg. Each time the ball hits a peg, it bounces to the right with probability p and to the left with probability 1p, independently from peg to peg. 1. What is the probability that when a ball reaches the bottom it ends up in the middle? How does this probability behave for large N for the special case that p = 1/2? 2. Suppose that M balls are dropped successively such that the balls do not encounter one another. How will the balls be distributed at the bottom of the board?