Solution
Fishing Until the Fourth Tag
Show the problem again
A lake contains N fish, 30 of which are tagged. Fish are caught one at a time, at random without replacement, until the fourth tagged fish appears. The fourth tagged fish is the 25th fish caught. What is the maximum likelihood estimate of N? The maximizer is unique.
Worked solution
The event is: among the first 24 catches exactly 3 are tagged, and the 25th is tagged. Its probability is L(N) = [C(30,3) C(N−30, 21) / C(N, 24)] · 27/(N−24). Taking the ratio of consecutive values and simplifying gives L(N)/L(N−1) = (N−30)(N−25) / (N(N−51)), which is at least 1 exactly when (N−30)(N−25) ≥ N(N−51), i.e. 750 ≥ 4N, i.e. N ≤ 187.5. Hence the likelihood increases up to N = 187 and decreases after, so N̂ = 187 = ⌊30·25/4⌋. The factor 27/(N−24) is the probability that the last fish is tagged; it must be included, since otherwise the stopping rule is ignored and you would be computing the wrong likelihood.
Source: Inverse-sampling (negative hypergeometric) capture-recapture design. Statement written for AxiomIQ.