Consider monitoring the number of critical cracks in the I-77 Interstate Highway starting at Columbia, SC (Mile Marker 0). Assume that the number of critical cracks could be modeled according to a homogeneous Poisson process (HHP) with rate \Lambda=.5 per mile [that is, on average, there are 0.5 critical cracks in a span of one mile of I-77].
(a) What is the probability that in the first mile of I-77 there will be zero critical cracks?
(b) What is the probability that in a span of I-77 covering 10 miles, you will find no more than 3 critical cracks?
(c) Starting at Mile Marker 0 in Columbia, SC, what is the probability that the first critical crack will be found on or before the fifth mile.
(d) On the average, how many miles, starting from Mile Marker 0, do you have to examine before you will find the first critical crack? [Note: At the point this is a challeng problem.]