Question

You Try! A wildlife conservation group wants to estimate the total number of fish in a large lake. They catch and tag 200 fish, then release them back into the lake. A few days later, they catch a sample of 500 fish and find that 40 of them are tagged.

Estimate the total fish population in the lake.

Answer 1

To estimate the total fish population in the lake, we can use the mark-recapture method. The basic idea is to use the ratio of tagged fish in the second sample to the total number of fish in that sample to estimate the total fish population in the lake.

Let:
- \( N \) = total estimated fish population in the lake
- \( M \) = number of tagged fish initially released = 200
- \( n \) = number of fish caught in the second sample = 500
- \( m \) = number of tagged fish in the second sample = 40

We set up the proportion:

\[
\frac{m}{n} = \frac{M}{N}
\]

Substituting the values we have:

\[
\frac{40}{500} = \frac{200}{N}
\]

Cross-multiplying gives us:

\[
40N = 500 \times 200
\]

Calculating the right side:

\[
500 \times 200 = 100000
\]

So we have:

\[
40N = 100000
\]

Next, we divide both sides by 40 to solve for \( N \):

\[
N = \frac{100000}{40} = 2500
\]

Therefore, the estimated total fish population in the lake is **2,500 fish**.