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Capture-Recapture

HigherHigher tier onlyAQAEdexcelOCR

Master capture-recapture for GCSE Maths with structured, exam-style practice. This Higher resource covers the capture-recapture method and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Assume the proportion of tagged animals matches the whole population.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

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WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Capture-recapture estimates the size of a population that cannot be counted directly, such as fish in a lake.

A sample is caught, marked and released. Later a second sample is caught, and the proportion of it that is marked estimates the proportion of the whole population that was marked.

The formula follows from that proportion: population \(= \dfrac{\text{first sample} \times \text{second sample}}{\text{number marked in second}}\). Several assumptions are needed, and questions often ask you to state them.

Revision notes

The method

Catch and mark a first sample, release it, then catch a second sample later.

Count how many of the second sample carry marks. That proportion should match the proportion of the whole population that was marked.

The formula

Population \(= \dfrac{n_1 \times n_2}{m}\), where \(n_1\) is the first sample, \(n_2\) the second, and \(m\) the marked ones recaptured.

If \(60\) are marked, \(80\) are caught later and \(12\) are marked, the estimate is \(\frac{60 \times 80}{12} = 400\).

The assumptions

The population has not changed between samples, the marks have not worn off, and the second sample is random.

If any assumption fails, the estimate is unreliable. Stating them is usually worth a mark.

Key points

  • Capture-recapture estimates an uncountable population.
  • Mark a first sample and release it.
  • Catch a second sample and count the marked ones.
  • Population \(= \frac{n_1 \times n_2}{m}\).
  • The population must not change between samples.
  • The second sample must be random.

Worked examples

Example 1

\(40\) fish are marked. Later \(50\) are caught, of which \(8\) are marked. Estimate the population.

Working

\[\frac{40 \times 50}{8}\]apply the capture-recapture formula
\[= \frac{2000}{8}\]work out the numerator
\[= 250 \text{ fish}\]state the estimate

Example 2

\(25\) birds are marked. Later \(30\) are caught, of which \(5\) are marked. Estimate the population.

Working

\[\frac{25 \times 30}{5}\]apply the formula
\[= 150 \text{ birds}\]state the estimate

Example 3

State two assumptions made in capture-recapture.

Working

\[\text{The population has not changed}\]no births, deaths or migration
\[\text{The second sample is random}\]and the marks have not worn off

Common mistakes

  • Dividing by the wrong figure.

    The denominator is the number of marked individuals in the second sample.

  • Forgetting to state the assumptions.

    Questions frequently award a mark for these.

  • Treating the answer as exact.

    It is an estimate, dependent on the assumptions holding.

  • Using the second sample size as the denominator.

    The denominator is the marked recaptures, not the whole second sample.

Exam tips

  • Write the formula out before substituting.
  • Identify which figure is the marked recapture count.
  • Describe the answer as an estimate.
  • Learn the three standard assumptions.

Key terms

Capture-recapture
A method of estimating population size.
Sample
A group caught from the population.
Marked
Tagged in the first sample.
Assumption
A condition that must hold for the estimate to be valid.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.