Problem 5.2. Three boys, A, B, and C, stand in a circle and play catch (B stands to the right of A). Before throwing the ball, each boy flips a coin to decide whether to throw to the boy on his right or left. If "heads" comes up, the boy throws to his right. If "trials" comes up, he throws to his left. The coin of boy A is "fair" (50% heads and 50% tails), the coin of boy B has heads on both side, and the coin of boy C is weighted (75% heads and 25% tails).

(a) Compute the transition matrix, its eigenvalues, and its left and right eigenvectors.

(b) If the ball is thrown at regular intervals, approximately what fraction of time does each boy have the ball (assuming they throw the ball many times)?

(c) If boy A has the ball to begin with, what is the chance he will have it after two throws? What is the chance he will have it after s throws?