Capture-Recapture

Capture-recapture estimates a population by marking a first sample, then checking how many marked items appear in a second sample. The key idea is that the fraction marked in the whole population should be about the same as the fraction marked in the second sample.

Common mistake: Keep the two fractions matched in the same order. For example, \(\dfrac{\text{marked first}}{\text{total population}}=\dfrac{\text{marked in second catch}}{\text{second catch total}}\).

Worked example:
In the first catch, 40 fish are tagged. In the second catch of 50 fish, 8 are tagged. Estimate the number of fish.
  1. Let \(x\) be the estimated population. Match the fraction marked in the whole population to the fraction marked in the second catch.
    \[\dfrac{40}{x}=\dfrac{8}{50}\]
    \[8x=40\times 50\]
    \[8x=2000\]
    \[x=250\]
Answer:
The estimated population is \(250\) fish.
Worked example:
A first sample has 36 deer marked. In a second sample of 70 deer, \(m\) are marked. The estimated population is 280. Calculate \(m\).
  1. Match the fraction marked in the whole population to the fraction marked in the second sample.
    \[\dfrac{36}{280}=\dfrac{m}{70}\]
    \[280m=36\times 70\]
    \[280m=2520\]
    \[m=9\]
Answer:
The number marked in the second sample is \(9\).
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