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GCSE Maths revision · Higher: probability and statistics

Capture-recapture

Higher only

The idea

To estimate a population: mark a sample, release it, then take a second sample. The fraction marked in the second sample estimates the fraction marked in the whole population.

  1. Set the two fractions equal:marked in sample 2size of sample 2=number first markedN\frac{\text{marked in sample 2}}{\text{size of sample 2}} = \frac{\text{number first marked}}{N}
  2. Solve for NN.
  3. State the assumptions: the marked ones mix in, and every animal is equally likely to be caught.

Watch out The estimate assumes the population did not change between samples.

See it worked

Worked exampleNon-calculator2 marks

Scientists catch 40 fish from a lake, tag them and release them. A week later they catch 50 fish; 8 of them are tagged.

Work out an estimate for the number of fish in the lake.

  1. The proportion tagged in the sample estimates the proportion in the lake.
  2.  850=40N  ⇒  N=40×508=250\frac{8}{50} = \frac{40}{N} \;\Rightarrow\; N = \frac{40 \times 50}{8} = 250

Answer 250

Your turn

Question 1Non-calculator3 marks

A capture-recapture method is used to estimate the number of rabbits in a field.

(a) State two assumptions the method relies on.

(b) Some tags fall off before the second sample. Explain the effect on the estimate.

Write your answer, then show the working
  1. (a) No rabbits are born, die, arrive or leave between the two catches.
  2. Tagged rabbits mix evenly and are as likely to be caught as untagged ones; tags are not lost.
  3. (b) N=tagged×sampletagged in sampleN = \frac{\text{tagged} \times \text{sample}}{\text{tagged in sample}}: a smaller denominator makes NN larger.

Answer (a) The population does not change between the samples, and every rabbit is equally likely to be caught (tagged rabbits mix back in).; (b) Fewer tagged rabbits are recognised, so the estimate is too high.

Question 2Non-calculator5 marks

A capture-recapture survey marks 40 insects. In a second sample of 50, 9 are recognised as marked.

(a) Estimate the population. Give your answer to the nearest whole number.

(b) A proposed estimate is twice your unrounded model estimate. What recapture count would produce it with the same sample sizes?

(c) If this required recapture count is not an integer, what does that mean?

Write your answer, then show the working
  1. (a) Use marked first sample times second sample divided by marked recaptures. 40×50/940\times 50/9
  2. Therefore 222222.
  3. (b) The estimate varies inversely with the marked recapture count. 9/29/2
  4. Therefore 92\frac{9}{2}.
  5. (c) With fixed sample sizes, no actual integer recapture count could give exactly that proposed doubled estimate.

Answer (a) 222222; (b) 92\frac{9}{2}; (c) With fixed sample sizes, no actual integer recapture count could give exactly that proposed doubled estimate.

Question 3Calculator allowed3 marks

Ali catches 30 fish from a pond, marks them and puts them back. Later she catches 40 fish, and 6 of them are marked. Estimate the number of fish in the pond.

In past papers

4 official questions in 4 papers since 2020. The latest:

All GCSE Maths past papers

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