Barbara and Dianne go target shooting. Suppose that each of...

Barbara and Dianne go target shooting. Suppose that each of Barbara's shots hits the wooden duck target with probability $p_{1}$, while each shot of Dianne's hits it with probability $p_{2}$. Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it was hit), what is the probability that (a) both shots hit the duck; (b) Barbara's shot hit the duck? What independence assumptions have you made?

Barbara and Dianne go target shooting. Suppose that each of Barbara's shots hits the wooden duck target with probability $p_{1}$, while each shot of Dianne's hits it with probability $p_{2}$. Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it was hit), what is the probability that
(a) both shots hit the duck;
(b) Barbara's shot hit the duck?
What independence assumptions have you made?

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Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the given condition, making it crucial for solving problems where outcomes depend on certain observed events.

Independence

Independence between events means that the occurrence of one event does not affect the occurrence of the other. In probability calculations, this assumption allows us to multiply the probabilities of individual events to find the joint probability, simplifying the analysis of simultaneous events.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is used to correctly calculate the probability of the union of two events by adding their individual probabilities and then subtracting the probability of their intersection. This ensures that outcomes that are common to both events are not counted twice.

Joint Probability

Joint probability refers to the probability of two events occurring simultaneously. When events are independent, the joint probability is the product of their individual probabilities, which is essential when evaluating scenarios with multiple simultaneous outcomes.

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