A certain system can experience three different types of defects. Let Ai (i = 1,2,3) denote the event that the system has a defect of type i. Suppose that the following probabilities are true. P(A1) = 0.10, P(A2) = 0.08, P(A3) = 0.05, P(A1 ∪ A2) = 0.12, P(A1 ∪ A3) = 0.12, P(A2 ∪ A3) = 0.11, P(A1 ∩ A2 ∩ A3) = 0.01
Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect? .
I know that I am basically looking for Pr[Exactly one] / Pr[at least one]. I know how to find Pr[at least one] which is basically, P(A1 ∪ A2 ∪ A3)= P(A1)+P(A2)+P(A3)-P(A1∩ A2)-P(A1∩ A3)-P(A2 ∩ A3)+ P(A1 ∩ A2 ∩ A3) = .37. I don't know how to find Pr[exactly one]. I believe Pr[eaxctly one] is equal to P(A1 ∩ A2' ∩ A3') +P(A1' ∩ A2∩ A3') +P(A1' ∩ A2' ∩ A3). But I do not know how find those probabilities