A pipeline is to be inspected. There is a 10% chance that any 100ft section of p
ID: 3259121 • Letter: A
Question
A pipeline is to be inspected. There is a 10% chance that any 100ft section of pipeline will contain at least one defect (thinning walls, weld defect, etc). Assume the sections are independent (whether one sections fails or not is independent of whether other sections fail or not).
What is the probability that, given 2 defective sections have been found so far, that exactly 15 sections of pipeline have been inspected?
What is the probability that, given 50 sections have been inspected so far, that fewer than 6 defective sections have been found?
What is the probability that the first defect will be found in the 10th pipeline inspected?
What is the average (expected) number of defects per 100 pipelines sections inspected?
Comment on the independence assumption. Do you believe this is a realistic assumption for a pipeline? Why or why not?
Explanation / Answer
p = 0.1
q = 0.9
P(exactly 15 sections of pipeline have been inspected | 2 defective sections have been found so far)
= 15C2 x 0.12 x0.913
= 0.2669
P(fewer than 6 defective sections have been found | 50 sections have been inspected so far) = P(X < 6)
= P(0) + P(1) + P(2) + P(3) + P(4) + P(5)
= 0.950 + 50x0.1x0.949 + 50C2x0.12x0.948 + 50C3x0.13x0.947 + 50C4x0.14x0.946 + 50C5x0.15x0.945
= 0.0051 + 0.0286 + 0.0779 + 0.1386 + 0.1809 + 0.1849
= 0.616
P(first defect will be found in the 10th pipeline inspected) = P(first 9 are non defective and 10th is defective)
= 0.99 x 0.1
= 0.0387
Average (expected) number of defects per 100 pipelines sections inspected = 100x0.1 = 10
The independednt assumption is not realistic as pipeline is a continous system with similar characteristics. So, the causative factor of defect on one section will have the same effect on other sections . Also, defects in one section can cause defects in other sections as well. So, this assumption is not realistic.
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