1. Bob, Kevin, and Jack are playing the following game: a single player will toss 5 fair coins. Assume the tosses are independent , and let H = 1 and T 0. Before the coin tosses, without consulting each other, each player chooses some number of the tosses (at least one) and calculates the sum (mod 2) of the tosses. That is the player’s score for example, supposed Kevin chose the second, third and fourth tosses, and Jack chose the first, second, and last toss. If the following coin tossing sequence occurs: HTHHT then Kevin’s score is 0 and Jack’s score is 1. Assume that each player does not select exactly the same subset of tosses.
(a) Show that the scores of the three players are pairwise independent
(b) Show that the scores of Bob, Kevin, and Jack are not always mutually independent. (A counter example when they are not independent is sufficient, but obviously explain your answer.)