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67 51.a.Let A= child has a food allergy, and R= child has a history of severe reaction. We are told that P(A) =.08 and P(R| A) = .39. By the multiplication rule, P(A∩R) = P(A) × P(R| A) = (.08)(.39) = .0312.b.Let M= the child is allergic to multiple foods. We are told that P(M| A) = .30, and the goal is to findP(M). But notice that Mis actually a subset of A: you can’t have multiple food allergies withouthaving at least one such allergy! So, apply the multiplication rule again:P(M) = P(M∩A) = P(A) × P(M | A) = (.08)(.30) = .024.52.We know that P(A1∪A2) = .07 and P(A1∩A2) = .01, and that P(A1) = P(A2) because the pumps areidentical. There are two solution methods. The first doesn’t require explicit reference to qor r: Let A1bethe event that #1 fails and A2be the event that #2 fails.Apply the addition rule: P(A1∪A2) = P(A1) + P(A2) – P(A1∩A2) ⇒.07 = 2P(A1) – .01 ⇒P(A1) = .04.Otherwise, we assume that P(A1) = P(A2) = qand that P(A1| A2) = P(A2| A1) = r(the goal is to find q).Proceed as follows: .01 = P(A1∩A2) = P(A1) P(