A and B are involved in a duel. The rules of the duel are...

$A$ 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 $p_{A},$ and each shot of $B$ will hit $A$ with probability $p_{B}$. 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 $n$ th round of shots? (d) the conditional probability that the duel ends after the $n$ th 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?

$A$ 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 $p_{A},$ and each shot of $B$ will hit $A$ with probability $p_{B}$. 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 $n$ th round of shots?
(d) the conditional probability that the duel ends after the $n$ th 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?

Key Concepts

Independent Events

This concept refers to situations where the outcome of one event does not affect the outcome of another. In the context of this problem, each shot taken by A or B is independent, meaning that the result of any given shot does not influence any other shot in subsequent rounds.

Bernoulli Trials

A Bernoulli trial is an experiment that results in a success with probability p and a failure with probability 1-p. In the duel problem, every round of shooting can be thought of as a Bernoulli trial for each shooter, where a 'hit' is a success and a 'miss' is a failure.

Geometric Distribution

The geometric distribution models the number of independent Bernoulli trials needed to obtain the first success. In this duel problem, multiple rounds are conducted until at least one hit occurs, making the number of rounds until the duel ends follow a geometric pattern.

Joint Probability

Joint probability concerns the likelihood of two (or more) events occurring together. For example, computing the probability that both duelists are hit in the same round requires evaluating the product of their individual probabilities, since the shots are independent.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. In this scenario, it is used to analyze outcomes based on additional information, such as the duel ending on a particular round provided that a certain participant was not hit.

Complementary Events

The concept of complementary events involves calculating the probability of an event not occurring by subtracting the probability of the event from one. This is used in the problem to determine probabilities like A not being hit when only the chance of getting hit is given.

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