A pumping station at a hydroelectric plant operates two identical pumps. Each pump has a probability of failure of 0.06, and the probability that both pumps fail is 0.0036. (a) Are failures in the two pumps mutually exclusive? b) What is the probability that at least one of the pumps fails? (c) Are failures in the two pumps independent?

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A pumping station at a hydroelectric plant operates two identical pumps. Each pump has a probability of failure of 0.06, and the probability that both pumps fail is 0.0036. (a) Are failures in the two pumps mutually exclusive? b) What is the probability that at least one of the pumps fails? (c) Are failures in the two pumps independent?

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a. P(A and B) = 0.0036 Check the conditional probability: P(A|B) = P(A and B)/P(B) (0.0036)/(0.06) does not equal P(A and B). Therefore, they are not mutually exclusive. b. P(At least one fails) = P(A or B fails) P(A or B failing) = P(A) + P(B) - P(A and B) = (0.06) + (0.06) - (0.06*0.06) c. Yes, they are independent of each other because (0.06)(0.06) = 0.0036, which is the same as P(A and B), and is the definition of independent events.
[a] No [b] Probability of all not failing = 0.94*0.94=(1-0.06)^2 = 0.8836 Probability of at least one fails = 1-0.8836 = 0.1164 [c] Yes

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