A certain system can experience three different types of defects. Let A i ( i =

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) = 0.12      P(A2) = 0.07     P(A3) = 0.05
P(A1 A2) = 0.13      P(A1 A3) = 0.14
P(A2 A3) = 0.10      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.)


(d) Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect? (Round your answer to four decimal place

Explanation / Answer

as P(A n B) =P(A)+P(B)-P(AnB)

hence

c)

probability of at least one defect =P(A U B U C)

probability of exactly one defect:

hence  probability that it has exactly one type of defect given system has at least one type of defect

=0.05/0.14 =0.3571

d)

probability that it does not have the third type of defect given  both of the first two types of defects

=P((A1 nA2)nA3c|(A1nA2)) =(P(A1 nA2)-P(A1 nA2 nA3))/P(A1nA2) =(0.06-0.01)/0.06 =0.8333

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