A capture-recapture model is a technique to estimate an unknown population by capturing, tagging, and re-capturing samples from the population.

In the article How many Mechanical Turk workers are there?, Panos Ipeirotis explains a simple version of a capture-recapture model as follows:

The simplest possible technique is the following:

• Capture/marking phase: Capture $n_1$ animals, mark them, and release them back.

• Recapture phase: A few days later, capture $n_2$ animals. Assuming there are $N$ animals overall, $n_1/N$ of them are marked. So, for each of the $n_2$ captured animals, the probability that the animal is marked is $n_1/N$ (from the capture/marking phase).

• Calculation: On expectation, we expect to see $n_2\cdot \frac{n_1}{N}$ marked animals in the recapture phase. (Notice that we do not know $N$.) So, if we actually see $m$ marked animals during the recapture phase, we set $m=n_2\cdot \frac{n_1}{N}$ and we get the estimate that:

$$N=n_1\cdot \frac{n_2}{m}$$

He adds that this basic version of a capture-recapture model makes the following assumptions, and the estimate $N$ can be inaccurate when these assumptions are violated:

• Assumption of no arrivals / departures (“closed population”): The vanilla capture-recapture scheme assumes that there are no arrivals or departures of workers between the capture and recapture phase.

• Assumption of no selection bias (“equal catchability”): The vanilla capture-recapture scheme assumes that every worker in the population is equally likely to be captured.