Chapter 14, Problem 74E

Darroch's method. is a method for estimating the size of a population using multiple (more than two) captures. For example, suppose that there are four captures of sizes $n_1,n_2,n_3,\text{and}{n}_4$, respectively, and let M be the total number of distinct individuals caught in the four captures (i.e., an individual that is captured in more than one capture is counted only once). Darroch's method gives the estimate for N as the unique solution of the equation $\left(1-\frac{M}{N}\right)=\left(1-\frac{n_1}{N}\right)\left(1-\frac{n_2}{N}\right)\left(1-\frac{n_3}{N}\right)\left(1-\frac{n_4}{N}\right)$.

a. Suppose that we are estimating the size of a population of fish in a pond using four separate captures. The sizes of the captures are $n_1=30,n_2=15,n_3=22,\text{and}{n}_4=45$. The number of distinct fish caught is $M=75$. Estimate the size of the population using Darroch's formula.

b. Show that with just two captures Darroch's method gives the same answer as the capture-recapture method.