Fifty Tagged Fish and a Tie
A lake contains an unknown number N of fish. Fifty are caught, tagged and released. After mixing, a second sample of 40 fish is drawn at random without replacement, and 8 of them carry tags. Treat the number of tagged fish in the second sample as hypergeometric. The likelihood of the data, as a function of the integer N, attains its maximum at exactly two values of N. What is the smaller of the two?
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Source: Lincoln-Petersen capture-recapture setting; the exact-likelihood tie is a standard remark in the literature. Statement written for AxiomIQ.