An aircraft seam requires 23 rivets The seam will have to be

An aircraft seam requires 23 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. (Round your answers to four decimal places.)

(a) If 22% of all seams need reworking, what is the probability that a rivet is defective? Correct: Your answer is correct.

(b) How small should the probability of a defective rivet be to ensure that only 12% of all seams need reworking?

Solution

Note: Shripal\'s answer to part (a) is wrong, because 1-.989804^23 = .21 (not .22).

Answers:

(a) .01074
(b) .00554

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Solutions/Discussion:

(a)
P(needs reworking) = P(at least 1 bad seam)

** P(at least 1) = 1 - P(0)

P(needs reworking) = P(at least 1 bad seam)
P(needs reworking) = 1 - P(0 bad seams)
P(needs reworking) = 1 - P(all good seams)
.22 = 1 - p^23
p^23 = 1 - .22
p^23 = .78

** take the 23rd root of both sides

(p^23)^(1/23) = .78^(1/23)
p = .98925547147875 <-- probability of a good seam
1-p = 1-.98925547147875
1-p = .01074452852125 <-- probability of a bad seam

Answer: .01074

Note: Let\'s check this answer: 1-.98926^23 = .22 Yep! :)

(b)
.12 = 1 - p^23
p^23 = 1 - .12
p^23 = .88
(p^23)^(1/23) = .88^(1/23)
p = .99445744428022
1-p = 1-.99445744428022
1-p = .00554255571978

Answer: .00554

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I hope this helped. If you have any questions, please ask them in the comment section. :)