An estate manager is responsible for stocking a small lake with fish. He begins by introducing 1000 fish into the lake and monitors their population growth to determine the likely carrying capacity of the lake. After one year an accurate assessment of the number of fish in the lake is taken and it is found to be 1200. Let N be the number of fish t years after the fish have been introduced to the lake.
Initially it is assumed that the rate of increase of N will be constant.
When t=8 the estate manager again decides to estimate the number of fish in the lake. To do this he first catches 300 fish and marks them, so they can be recognized if caught again. These fish are then released back into the lake. A few days later he catches another 300 fish, releasing each fish after it has been checked, and finds 45 of them are marked.
Assuming the proportion of marked fish in the second sample is equal to the proportion of marked fish in the lake. Let X be the number of marked fish caught in the second sample, where X is considered to be
distributed as B(n,p).