Poisson Process and Stress Fractures in Railway Lines

[Jenny123]

Homework Statement

Suppose that stress fractures appear in railway lines according to a Poisson process at a rate of 2 per month.
a)Find the probability that the 4th stress fracture on the railway line occurred 3 months after the process of checking the new railway lines.

b)Suppose new railway lines have just been laid, how long (in months) is it expected to take for the 4th stress factor to occur?

Homework Equations

The Attempt at a Solution

a) P(N(3) =3) = (54 exp{-6} )/3!

b) P(N(4) =4) = (512 exp{-8})/4!

 

[chemisttree]

What is your formula for the probability function? I don't see how you get 54 in part a.
You appear to use the same probability estimation to determine the number of months in b. Is there a confidence level for b?