Problem4 (20 points; 2pnts each) A seawall will develop large but repairable cra

Question

Problem4 (20 points; 2pnts each) A seawall will develop large but repairable cracks on the average of once every 15 years. When the 8th crack occurs, the seawall must be replaced. Assume the occurrence of cracks is an independent Bernoulli seq uence a) How long is the seawall expected to last? b) What is the expected number of cracks in 100 years? (100 years is the hopeful serviceable life for many large structures.) c) What is the probability that the seawall will last at least 100 years? (100 years is the hopeful serviceable life for many large structures.) Full points will be awarded writing out the formula with numbers. d) What is the probability of 2 cracks in 15 years? e) What is the probability that the first crack will occur during the first 15 years of operation? Full points will be awarded for writing out the formula with number.s. f) When is the first crack expected? g) What is the standard deviation for when the first crack is expected to appear? h) What is the probability that the first crack will occur during Year 15 of operation? i) What is the probability that a crack will occur during Year 15 of operation?

Explanation / Answer

(a) Time seawall is expected to last = 15 * 8 = 120 years

(b) Expected number of cracks in 100 year = 100/ 15 = 6.67 cracks

(c) Probability that seawall will last 100 years) = ?

NUmber of cracks per year = 1/15 = 0.0667 cracks

Expected number of cracks in 100 years = 100 * 0.0667 = 6.667

so probability of occuring less than 8 cracks in seawall in 100 years when expected number of cracks are 6.667

We will apply poisson approximation of binomial as p < 0.01 and n >= 100formula.

Let see X is the number of cracks occured

Pr( X<8; = 6.667) = POISSON ( Cracks < 8 ; 6.6667) = e- X/X! = 0.6482

(d) Expected number of cracks in 15 years = 1

so Pr( X = 2; 1) = BIN (X =2; 15; 0.0667) = 0.1903

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