In each of the problems below, show the different stages of mathematical modelling for solving the problems. 1. An ornithologist wants to estimate the number of parrots in a large field. She uses a net to catch some, and catches 32 parrots, which she rings and sets free. The following week she manages to net 40 parrots, of which 8 are ringed. (i) What fraction of her second catch is ringed? (ii) Find an estimate of the total number of parrots in the field.

Understand the Question The capture-recapture method estimates an unknown total population () using proportions. Initial capture: parrots are caught, ringed, and released into the population. Second capture: parrots are caught, out of which are ringed. Assuming the proportion of ringed parrots in the sample equals the proportion in the entire population: Part (i) What fraction of her second catch is ringed? Step 1 · Calculate the Fraction of Ringed Parrots From the second catch: Number of ringed parrots Total parrots caught Answer > (i) Part (ii) Find an estimate of the total number of parrots in the field. Step 1 · Estimate the Total Population Let be the total estimated number of parrots in the field. Assuming the proportion of ringed parrots in the sample equals that of the whole population: Answer > (ii) Common Mistakes Inverting the Proportion: Writing on one side and on the other. Always keep corresponding terms in the numerator and denominator consistent. Direct Addition: Simply adding both catches () instead of using the capture-recapture proportion to estimate the total population.

Appendix 2: Mathematical Modelling | A2.2

Question 1

In each of the problems below, show the different stages of mathematical modelling for solving the problems.

An ornithologist wants to estimate the number of parrots in a large field. She uses a net to catch some, and catches 32 parrots, which she rings and sets free. The following week she manages to net 40 parrots, of which 8 are ringed.

(i) What fraction of her second catch is ringed? (ii) Find an estimate of the total number of parrots in the field.

Solution

Understand the Question

The capture-recapture method estimates an unknown total population ( $N$ ) using proportions.
Initial capture: $32$ parrots are caught, ringed, and released into the population.
Second capture: $40$ parrots are caught, out of which $8$ are ringed.
Assuming the proportion of ringed parrots in the sample equals the proportion in the entire population:

$\dfrac{\text{Ringed in second catch}}{\text{Total in second catch}} = \dfrac{\text{Total ringed initially}}{\text{Total population (} N \text{)}}$

(i) What fraction of her second catch is ringed?

Step 1 · Calculate the Fraction of Ringed Parrots

From the second catch:

Number of ringed parrots $= 8$
Total parrots caught $= 40$

\begin{aligned} \text{Fraction} &= \dfrac{\text{Number of ringed parrots}}{\text{Total parrots caught}} \\[0.6em] &= \dfrac{8}{40} \\[0.6em] &= \dfrac{1}{5} \end{aligned}

Answer

(i) $\dfrac{1}{5}$

(ii) Find an estimate of the total number of parrots in the field.

Step 1 · Estimate the Total Population

Let $N$ be the total estimated number of parrots in the field.

Assuming the proportion of ringed parrots in the sample equals that of the whole population:

\begin{aligned} \dfrac{\text{Ringed in second catch}}{\text{Total in second catch}} &= \dfrac{\text{Total ringed initially}}{\text{Total population}} \\[0.6em] \dfrac{8}{40} &= \dfrac{32}{N} \\[0.6em] \dfrac{1}{5} &= \dfrac{32}{N} \\[0.6em] N &= 32 \times 5 \\[0.6em] N &= 160 \end{aligned}

Answer

(ii) $160$

Common Mistakes

Inverting the Proportion: Writing $\dfrac{\text{Total}}{\text{Ringed}}$ on one side and $\dfrac{\text{Ringed}}{\text{Total}}$ on the other. Always keep corresponding terms in the numerator and denominator consistent.
Direct Addition: Simply adding both catches ( $32 + 40 = 72$ ) instead of using the capture-recapture proportion to estimate the total population.

Question 1

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