Question

The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of 3 cracks per mile. Round your answers to four decimal places (e.g. 98.7654). (a) What is the probability that there are no cracks that require repair in 22 miles of highway

Answer 1

0.0025 = 0.25% probability that there are no cracks that require repair in 2 miles of highway.

Explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\mu` is the mean in the given time interval.

Mean of 3 cracks per mile

In this problem, we are going to calculate a probability in 2 miles. This means that
`\mu = 2*3 = 6`

(a) What is the probability that there are no cracks that require repair in 2 miles of highway

This is P(X = 0). So

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

`P(X = 0) = (e^(-6)*(6)^(0))/((0)!) = 0.0025`

0.0025 = 0.25% probability that there are no cracks that require repair in 2 miles of highway.