Conditional Probability- Defect Question

Question

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

Answer 1

Draw a Venn diagram like this with circles for three sets $A_1$, $A_2$, and $A_3$. This divides the plane into 8 regions: the 1 region outside all three sets, the 3 regions inside just one set, the 3 regions inside two, but not all three sets, and the 1 region inside all three sets. Now, assume these three sets represent “has defect of type one,” “has defect of type two,” and has “defect of type three,” respectively.

All your probabilities are multiples of $0.01=\frac{1}{100}$, so place numbers — the numbers represent “percentage of systems” — totaling 100 in the different regions of the Venn diagram to represent the specific probabilities (out of the sum 100) in your problem. For example, $P(A_1)=0.10$, so the sum of the numbers in the four regions within $A_1$ will be $10$. You have enough information to figure out every one of the eight numbers in the Venn diagram.

Finally, look at the part of the region in at least one of the circles (seven of the eight regions). This is where you have at least one defect. Out of the total of the numbers here, what fraction of the numbers are in one of the three regions that represent a single defect (numbers not in any of the intersections of the $A_i$. That’s your answer.