Fish are caught at random according to a Poisson process at

Fish are caught at random according to a Poisson process at a rate of one fish per hour. If exactly three fish are ccaught in the first hour, what is the probability that it took more than one-half hour to catch the first fish?

Solution

If he catches the first fish after first half-hour, then he must have caught all 3 fishes in the second half hour.

For each of the 3 fishes caught, the probability that it was caught in the first half hour is 1/2

P(more than one half hour to catch the fish|3 fishes caught in the first 2 half hours))=(1/2)^3=0.125