Fish in the lake
To estimate the number of fish in a lake, 100 fish are caught, tagged and released. A week later, after they have mixed with the others, 50 fish are caught, and 10 of them carry a tag.
Estimate the number of fish in the lake by assuming that the tagged fraction of the second catch equals the tagged fraction of the whole lake (the Lincoln-Petersen estimate).
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
If the lake holds fish, what fraction of them is tagged?
Set that fraction equal to .
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The second catch is a random sample of the lake, so the share of tagged fish in it should be close to the share of tagged fish in the lake, which is .
Step 1. In the lake, of the fish are tagged. In the second catch, are.
Step 2. Matching the two:
About the method. This is also, up to a rounding detail, the maximum likelihood estimate: the number of tagged fish in the second catch is hypergeometric, and its likelihood in is largest at (here 499 and 500 tie). The estimate is biased upwards for small samples; Chapman's version corrects most of that. It assumes the population is closed and that tagged fish are as easy to catch as the others.