Bob, Eve and Carol bought new slingshots. Bob is not very accurate, hitting his target with prob- ability 1/3. Eve is better, hitting her target with probability 2/3. Carol never misses. They decide to play the following game: They take turns shooting each other. For the game to be fair, Bob starts first, then Eve and finally Carol. Any player who gets shot has to leave the game. The last person standing wins the game. In this problem, we will investigate what Bob's best course of action would be. (a) Compute the probability of the event Eị that Bob wins in a duel against Eve alone, assuming he shoots first. (Hint: Let x be the probability Bob wins in a duel against Eve alone, assuming he fires first. If Bob misses his first shot and then Eve misses her first shot, what is the probability Bob wins in terms of x?)
(b) Compute the probability of the event E2 that Bob wins in a duel against Eve alone, assuming he shoots second.
(c) Compute the probability of the same events for a duel of Bob against Carol. (d) Assuming that both Eve and Carol play rationally, conclude that Bob's best course of action is to shoot into the air (i. e., intentionally miss)! (Hint: What happens if Bob misses? What if he doesn't?)