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For an avian research study, scientists wanted to estimate the local robin population. The scientists captured, tagged, and released $84$ robins. A week later, they noted $42$ tagged robins out of the $380$ robins they counted. To the nearest whole number, what is the best estimate for the robin population?

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Estimate population size using proportions

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Q. For an avian research study, scientists wanted to estimate the local robin population. The scientists captured, tagged, and released

84

robins. A week later, they noted

42

tagged robins out of the

380

robins they counted. To the nearest whole number, what is the best estimate for the robin population?

Set up proportion: Step $1$ : Set up the proportion based on the capture-recapture method. $\newline$ We know: $\newline$ - Tagged robins found: $42$ $\newline$ - Total robins counted in second capture: $380$ $\newline$ - Total robins tagged initially: $84$ $\newline$ Let $R$ be the estimated robin population. $\newline$ The proportion is set as: $\newline$ $\frac{42}{380} = \frac{84}{R}$
Solve proportion: Step $2$ : Solve the proportion by cross-multiplying to find $R$ . Cross-multiplying gives: $42 \times R = 84 \times 380$
Continue solving: Step $3$ : Continue solving for $R$ . Divide both sides by $42$ to isolate $R$ : $R = (84 \times 380) / 42$ $R = 760$

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