Question

A fisherman catches fish in a river according to a Poisson process with a rate equal to 2 fish per hour. He starts fishing at 6 AM. (b) Find the probability that he will catch exactly one fish between 6 AM and.......

Answer 1

To find the probability that the fisherman will catch exactly one fish between 6 AM and a given time using the Poisson distribution, calculate the average number of fish caught in the given time interval and use the Poisson probability formula.

Step-by-step explanation:

To find the probability that the fisherman will catch exactly one fish between 6 AM and a given time, we can use the Poisson distribution. The average rate of catching fish is given as 2 fish per hour. So, for a given time interval, the average number of fish caught can be calculated by multiplying the average rate by the length of the time interval.

Let's say the time interval is t hours. Therefore, the average number of fish caught in t hours is 2t. Now, using the Poisson probability formula, we can calculate the probability of catching exactly one fish in this time interval:

P(X = 1) = (e^(-2t) * (2t)^1) / 1!

For example, if the time interval is 3 hours:

P(X = 1) = (e^(-2 * 3) * (2 * 3)^1) / 1!

This calculation will give you the probability that the fisherman will catch exactly one fish between 6 AM and the given time.