Nontransitive Dice

Three dice have the following faces. Die A: {2, 2, 4, 4, 9, 9}. Die B: {1, 1, 6, 6, 8, 8}. Die C: {3, 3, 5, 5, 7, 7}. Each die is fair (all six faces equally likely). Two players each choose a die and roll it; the higher number wins. (a) Compute $P(A > B)$, $P(B > C)$, and $P(C > A)$. (b) Show that these dice are nontransitive: A tends to beat B, B tends to beat C, yet C tends to beat A. (c) You are in a game where your opponent chooses a die first, then you choose. Which die should you pick in each case, and what is your winning probability?