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Chapter 23: Problem 1

\(A\) and \(B\) are independent events with \(P(A)=\) \(0.75\) and \(P(B)=0.9 .\) The compound event ' \(A\) occurs, then \(A\) occurs, then \(B\) occurs' is denoted by \(A A B\). Other compound events are denoted in a similar way. Calculate the probability of the following compound events occurring: (a) \(A A B\) (b) \(B B A\) (c) \(A A B B\)

Short Answer

Expert verified
Question: Calculate the probabilities of the following compound events: (a) AAB, (b) BBA, and (c) AABB. Answer: The probabilities of the compound events are as follows: (a) P(AAB) = 0.50625 (b) P(BBA) = 0.6075 (c) P(AABB) = 0.455625

Step by step solution

01

(a) Finding the probability of \(A A B\) happening

Since A and B are independent events, the probability of the compound event \(A A B\) is the product of their individual probabilities. Therefore, the probability of the event \(A A B\) happening is: \(P(AAB) = P(A) \times P(A) \times P(B) = 0.75 \times 0.75 \times 0.9\) Calculating: \(P(AAB) = (0.75)(0.75)(0.9) = 0.50625\)
02

(b) Finding the probability of \(B B A\) happening

The probability of the compound event \(B B A\) is the product of individual probabilities of events B, B, and A. Therefore, the probability of the event \(B B A\) happening is: \(P(BBA) = P(B) \times P(B) \times P(A) = 0.9 \times 0.9 \times 0.75\) Calculating: \(P(BBA) = (0.9)(0.9)(0.75) = 0.6075\)
03

(c) Finding the probability of \(A A B B\) happening

The probability of the compound event \(A A B B\) is the product of individual probabilities of events A, A, B, and B. Therefore, the probability of the event \(A A B B\) happening is: \(P(AABB) = P(A) \times P(A) \times P(B) \times P(B) = 0.75 \times 0.75 \times 0.9 \times 0.9\) Calculating: \(P(AABB) = (0.75)(0.75)(0.9)(0.9) = 0.455625\) In conclusion, the probabilities of the compound events are: (a) \(P(AAB) = 0.50625\) (b) \(P(BBA) = 0.6075\) (c) \(P(AABB) = 0.455625\)

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