Recently, there has been an epidemic among prairie dogs. Christine is a scientist who studies how diseases spread and wants to estimate the size of a particular prairie dog town. First she catches 85 prairie dogs and tags them. These prairie dogs are then released, and at a later date, 994 prairie dogs are caught. Among these prairie dogs, 17 are tagged. To the nearest whole number, what is the best estimate for the prairie dog population?

To estimate the total prairie dog population, we can use the **mark-and-recapture method**. The idea is that the proportion of tagged prairie dogs in the second sample should represent the proportion of the entire population that was tagged initially.

Let:
- \( N \) represent the total population of prairie dogs,
- \( 85 \) be the number of prairie dogs initially tagged,
- \( 994 \) be the number of prairie dogs caught in the second sample,
- \( 17 \) be the number of tagged prairie dogs in the second sample.

The proportion of tagged prairie dogs in the second sample should be equal to the proportion of the total population that was tagged:

\[
\frac{17}{994} = \frac{85}{N}
\]

Now, solve for \( N \):

\[
N = \frac{85 \times 994}{17}
\]

Calculating:

\[
N = \frac{84590}{17} \approx 4976.47
\]

To the nearest whole number, the best estimate for the prairie dog population is:

\[
N \approx 4976
\]

Thus, the estimated prairie dog population is **4,976**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the mark-and-recapture method account for potential bias in estimating populations?
2. What assumptions must hold true for the mark-and-recapture method to be accurate?
3. How would the estimate change if fewer tagged prairie dogs were found in the second sample?
4. What are other methods for estimating animal populations in the wild?
5. What would happen to the population estimate if more than 17 tagged prairie dogs were recaptured?

**Tip:** Ensure that tagging does not affect the animals' behavior or their likelihood of being caught again.

Learn how to estimate the population of prairie dogs using the mark-and-recapture method. This example involves tagging 85 prairie dogs, recapturing 994, and finding 17 tagged animals. Find out how proportions help estimate the total population of 4,976 prairie dogs.

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