Measurement, STD2 2026 HSC 15 MC

A scientist is estimating the population of a bee colony.

The scientist captures $p$ bees but only marks $k\%$ of them and releases them all back into the colony. The next day, the scientist captures $q$ bees and finds that $r$ bees were marked.

Assuming all bees have an equal chance of being captured, which expression gives the best estimate of the total bee population?

$\dfrac{kpq}{100r}$
$\dfrac{100r}{kpq}$
$\dfrac{q(p-k)}{100r}$
$\dfrac{r}{q}-\dfrac{kp}{100}$

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$A$

Show Worked Solution

$\text{Let $N$ = bee population}$

$\text{Capture} = \dfrac{\frac{k}{100}\times p}{N} = \dfrac{kp}{100N}$

$\text{Recapture} = \dfrac{r}{q}$

$\text{Since capture/recapture ratios are equal:}$

$\dfrac{kp}{100N}$	$=\dfrac{r}{q}$
$\dfrac{100N}{kp}$	$=\dfrac{q}{r}$
$N$	$=\dfrac{kpq}{100r}$

$\Rightarrow A$

\(\dfrac{kp}{100N}\)	\(=\dfrac{r}{q}\)
\(\dfrac{100N}{kp}\)	\(=\dfrac{q}{r}\)
\(N\)	\(=\dfrac{kpq}{100r}\)