Question 1

We will use the capture-recapture method to estimate the total parrot population.

Step 1 — Fraction of ringed parrots

Let's find the fraction of ringed parrots. We look at the second catch. Number of ringed parrots is 8. Total parrots caught are 40.

$\text{Fraction} = \frac{\text{Number of ringed parrots}}{\text{Total parrots caught}}$

$= \frac{8}{40}$

$= \frac{1}{5}$

$\boxed{\frac{1}{5}}$

Step 2 — Estimate total parrots

Let $N$ be the total number of parrots. We assume the proportion of ringed parrots is the same. Initially, 32 parrots were ringed. In the second catch, 8 out of 40 were ringed.

$\frac{\text{Ringed in second catch}}{\text{Total in second catch}} = \frac{\text{Total ringed initially}}{\text{Total population}}$

$\frac{8}{40} = \frac{32}{N}$

$\frac{1}{5} = \frac{32}{N}$

$N = 32 \times 5$

$\boxed{160 \text{ parrots}}$

Answer

(i) The fraction of her second catch that is ringed is $\frac{1}{5}$. (ii) An estimate of the total number of parrots in the field is 160.