The Petersen capture-recapture formula to estimate a population size; the assumptions the method relies on and their appropriateness; the role of sample size in the reliability of the estimate.

What this dot point is asking

Edexcel Higher tier code 2h.02 requires you to apply the Petersen capture-recapture formula to estimate the size of a population that cannot be counted directly, and to know the assumptions the method relies on and judge their appropriateness in practice. It is the classic technique for estimating wildlife populations, and it connects sampling, proportion and inference.

The capture-recapture idea

The logic is a proportion: if a known number of individuals are marked and released, then the fraction of the second sample that is marked should equal the fraction of the whole population that is marked. Turning that equality round gives the population estimate.

The Petersen formula

So for $60$ fish marked, a recapture of $80$ fish containing $12$ marked, $N = \frac{60 \times 80}{12} = 400$. Set up the proportion carefully (the marked fraction of the recapture equals the marked fraction of the population), then solve for $N$.

The assumptions

The estimate is only valid if the method's assumptions hold:

1. The population is closed: no births, deaths, immigration or emigration between the two samples.
2. Marked individuals mix back evenly with the rest of the population before the second sample.
3. The marks do not come off and do not change an individual's chance of being caught (or its survival).
4. Every individual is equally likely to be caught in each sample.

Edexcel expects you to state these assumptions and to comment on whether they are reasonable in a given context. For example, if marking makes fish easier for predators to catch, the closed-population and equal-catch assumptions break down, and the estimate becomes unreliable.

Sample size and reliability

As with all estimation, larger samples give a more reliable result. A bigger second sample contains more individuals, so the proportion of marked ones is estimated more accurately and random variation has less effect on $N$. A very small recapture (few marked individuals) makes the estimate volatile, because a change of one or two marked recaptures swings $N$ a long way.

You can see this sensitivity directly in the formula $N = \frac{M \times n}{m}$: because $m$ is in the denominator, a small $m$ makes $N$ change sharply if $m$ moves by even one. For instance, with $M = 60$ and $n = 80$, a recapture of $m = 12$ gives $N = 400$, but $m = 10$ gives $N = 480$ and $m = 15$ gives $N = 320$. Catching a larger second sample (so $m$ is larger) reduces this instability and narrows the likely range of the estimate.

Why capture-recapture is used

Capture-recapture matters because many populations cannot be counted directly: you cannot line up and count every fish in a lake, every bird in a wood or every insect in a field. By marking and recapturing, you turn an impossible census into a manageable pair of samples, then use proportion to infer the whole. The same idea is used in real ecology and conservation to monitor wildlife numbers over time. Understanding both how to apply the formula and when its assumptions make it trustworthy is what the qualification rewards, because a method is only as good as the assumptions behind it.