A certain system can experience three different types of def

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) = 0.06
P(A₁ A₂) = 0.12      P(A₁ A₃) = 0.13
P(A₂ A₃) = 0.12      P(A₁ A₂ A₃) = 0.01


(c) Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect? (Round your answer to four decimal places.)

Solution

Note that

P(A1 U A2 U A3) = P(A1) + P(A2) + P(A3) - P(A1 and A2) - P(A2 and A3) - P(A1 and A3) + P(A1 and A2 and A3)

and

P(A and B) = P(A) + P(B) - P(A U B)

Then

P(A1 and A2) = 0.06
P(A2 and A3) = 0.02
P(A1 and A3) = 0.03

Thus,

P(A1 U A2 U A3) = 0.14

Now, the probability that it has exactly 1 defect is

P(A1 only or A2 only or A3 only) = P(A1) + P(A2) + P(A3) - 2P(A1 and A2) - 2P(A2 and A3) - 2P(A1 and A3) + 3P(A1 and A2 and A3)

P(exactly 1) = P(A1 only or A2 only or A3 only) = 0.05

Thus,

P(exactly 1|at least 1) = P(exactly 1)/P(at least 1) = 0.05/0.14 = 0.357142857 [ANSWER]