Question: A and B are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at

A and B are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of A will hit B with probability pA, and each shot of B will hit A with probability pB. What is
(a) The probability that A is not hit?
(b) The probability that both duelists are hit?
(c) The probability that the duel ends after the nth round of shots?
(d) The conditional probability that the duel ends after the nth round of shots given that A is not hit?
(e) The conditional probability that the duel ends after the nth round of shots given that both duelists are hit?

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