PART 1 An engineering system has two components. Let us define thefollowing even

Question

PART 1 An engineering system has two components. Let us define thefollowing events:

Describe the following events in the above terms: (a) At least one of the components if good (b) One is good and one is defective
PART 2 Tests on these two components have produced the followingresults:

Determine the probability that: (a) The second component is good (b) At least one of the components is good (c) The first component is good given that the second isgood (d) The first component is good given that at most onecomponent is good
For the two events A and B: (e) Are they independent? Verify your answer. (f) Are they mutually exclusive? Verify your answer. PART 1 An engineering system has two components. Let us define thefollowing events: A: first component is good A bar: first component is defective B: second component is good B bar: second component is defective Describe the following events in the above terms: (a) At least one of the components if good (b) One is good and one is defective PART 2 Tests on these two components have produced the followingresults: P[A] = 0.8 P[B] = 0.85 P[B|A] bar = 0.75 Determine the probability that: (a) The second component is good (b) At least one of the components is good (c) The first component is good given that the second isgood (d) The first component is good given that at most onecomponent is good For the two events A and B: (e) Are they independent? Verify your answer. (f) Are they mutually exclusive? Verify your answer.

Explanation / Answer

Let A'= not event A. We are given P(A)=.8, P(B/A)=.85 and P(B/A')=.75 P(A')=.2, P(B'/A)=.15 a)P(AUB) b)P(AnB')+P(A'nB) Part 2 a)P(B)=P(B/A)P(A)+P(B/A')P(A')=(.85)(.8)+(.75)(.2)=.83   (Which also means P(B')=1-.83=.17) b)P(AUB)=P(A)+P(B)-P(AnB)=.8+.83-P(B/A)P(A)=.8+.83-.85(.8)=.95 c)P(A/B)={P(B/A)P(A)}/P(B)=(.85)(.8)/.83=.85 d)P(A/B')={P(B'/A)P(A)/P(B')}=.15(.8)/.17=.71 e)Independent means P(A/B)=P(A) .85 in not equal to .8hence A and B are not independent. f)If mutually exclusive thenP(AuB)=P(A)+P(B) In our caseP(AUB)=.95 which is not equal to P(A)+P(B)=.8+.83   Hence they are not mutuallyexclusive

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