Jack likes to go fishing. While waiting for the fish to bite, he formulates the following model for the process: fish bite according to a Poisson process with intensity 4 bites per hour. Biting fish are caught independently, and on average only one in two times.
My solutions so far:
N(2) = # of fish biting during 2 hours = Poisson(8)
M(2) = # of fish caught during two hours = Poisson(4)
a) What is the probability that six fish bite during the first two hours?
P(N(2) = 6) is what we have to solve for.
b) What is the probability that he fails to catch any fish during the first two hours?
P(M(2) = 0) = $e^{-4}$ = 0.018 = 1.8% chance he catches no fish during the first two hours.
c) What is the probability that, during the first two hours, six fish bite and two of these are caught?
P(M(2) = 2|N(2) = 6)
I need help solving part a and c as I am not sure how to properly solve for these probabilities. I am confident in my solution for part b. Thanks!