1. The number of cracks in a section of interstate highw ay that are significant

Question

1. The number of cracks in a section of interstate highw ay that are significant enough to require repair to folow a Poisson distribution with a mean of 2 cracks per mile. (a) What is the probability that there are no crack that require repai r in 5 miles of highway? (b) what is the probability that at least one crack requires repair in ½ mile of highway (c )Find the minimum number of cracks in 1/2 mile of highway so that the probability of this number or fewer cracks is at least 0.98. (d) if the number of cracks is related to vehicle load on the highway and some sections of the highway have a heavy load of vehicles whereas other sections carry a light load, what do yo think about the assumption of a Poisson distribution for the cracks that require repair?

Explanation / Answer

a)
mean = 2 cracks per mile
mean = 10 crack in 5 miles
P(X = 0) = e^(-10) = 0.0000453999
b)
mean = 1 crack per 1/2 mile
P(X > 0)
= 1 - P(X = 0)
= 1- 1/e = 0.6321205

c)
P(X<= n) >= 0.98
n = 3
P(X < = 3) = 0.9810
P(X < = 2) = 0.9196
hence
3 is correct

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