Final answer:
The probability of finding no cracks in 5 miles of highway is approximately 0.00454%. The probability of at least one crack in 1 mile is around 86.47%.
Step-by-step explanation:
The number of significant cracks per mile of interstate highway that require repair can be modeled by a Poisson distribution with a mean (λ) of 2 cracks per mile. To solve these problems, we use the properties of the Poisson distribution.
a. Probability of no cracks in 5 miles
To find the probability of no cracks in 5 miles, we scale our distribution for the total distance. If the mean is 2 per mile, for 5 miles the mean becomes 2 × 5 = 10 cracks. The probability of x cracks in a Poisson distribution is given by P(X = x) = (λx × e−λ) / x!.
So for no cracks, x = 0, and the formula becomes P(X = 0) = (100 × e− 10) / 0! = e− 10. This is approximately equal to 0.0000454 or 0.00454% probability.
b. Probability of at least one crack in 1 mile
The probability of at least one crack in a mile is the complement of the probability of no cracks. The formula for no cracks in a Poisson distribution with a mean of 2 cracks per mile is P(X = 0) = (20 × e− 2) / 0! = e− 2.
To find the complement, we subtract this value from 1: 1 - e− 2 ≈ 0.8647 or 86.47% probability.