Capture-Recapture Method: A small lake has 240 bass fish. A net is used to catch
ID: 2915978 • Letter: C
Question
Capture-Recapture Method: A small lake has 240 bass fish. A net is used to catch a sample of 30 bass fish. Each bass that is caught is marked with a tag and returned to the lake. A month later, a second sample of 24 bass is caught using a net and the number of bass with tags is noted. What is the distribution of the number of tagged bass in the second sample? What is the expected number of tagged fish in the second sample? Suppose that the number of bass in the lake is unknown and call the unknown number N . Suppose that exactly 5 tagged bass were found in the second sample. Use your answer in part (b) to obtain a guess at the value of N .Explanation / Answer
A) There are 30 tagged bass out of 240. Since we are taking fishthe 2nd time and counting them before replacing all of them, thisis a hypergeometric distribution. The probability densityfunction of a hypergeometric distribution is... P(X = k) = C(m, k) * C(N-m, n-k) where the population has size N,there are m possible successes, and the sample size is n. B) The mean of a hypergeometric distribution is n*m/N. Thus, theexpected number of tagged fish in the 2nd sample is 24 * 30 / 240 =3. C) We want to maximize the probability of having X = 5 by changingN. In other words, we want to adjust N such that the mode is5. The mode of a hypergeometric distribution is (n+1) * (m+1)/ (N+2). Thus, we want to solve 25 * 31 / (N+2) = 5 N = 153