Calculating the Proportion of Tagged Fish Recaptured

Oct 13, 2025 by leolavanderia.com.br 81 views

Introduction

In this article, we will explore a real-world problem in biology, where we need to estimate the number of fish in a pond based on a sample of 100 fish that were caught, tagged, and released back into the pond. We will use the data provided to calculate the best estimate of the number of fish in the pond.

The Data

The data provided consists of the number of fish caught in each of 16 sampling periods, along with the number of tagged fish recaptured in each period. The data is as follows:

Sampling Period	Number of Fish Caught	Number of Tagged Fish Recaptured
1	10	1
2	12	2
3	15	3
4	18	4
5	20	5
6	22	6
7	25	7
8	28	8
9	30	9
10	32	10
11	35	11
12	38	12
13	40	13
14	42	14
15	45	15
16	48	16

The Methodology

The method we will use to estimate the number of fish in the pond is called the "mark-release-recapture" method. This method involves catching a sample of fish, marking them with a distinctive tag, and then releasing them back into the pond. Later, a second sample of fish is caught, and the number of tagged fish recaptured is counted. By using the ratio of tagged fish recaptured to the total number of fish caught in the second sample, we can estimate the total number of fish in the pond.

Calculating the Estimate

To calculate the estimate, we need to use the following formula:

N = (M / m) * K

where N is the total number of fish in the pond, M is the number of tagged fish released, m is the number of tagged fish recaptured, and K is the total number of fish caught in the second sample.

However, in this case, we do not have the total number of fish caught in the second sample. Instead, we have the number of fish caught in each of 16 sampling periods. To calculate the estimate, we need to use the following formula:

N = (M / m) * (n / p)

where N is the total number of fish in the pond, M is the number of tagged fish released, m is the number of tagged fish recaptured, n is the number of fish caught in the second sample, and p is the proportion of tagged fish recaptured.

Calculating the Proportion of Tagged Fish Recaptured

To calculate the proportion of tagged fish recaptured, we need to divide the number of tagged fish recaptured by the total number of fish caught in the second sample. However, since we do not have the total number of fish caught in the second sample, we can use the following formula:

p = (m / M) * (n / N)

where p is the proportion of tagged fish recaptured, m is the number of tagged fish recaptured, M is the number of tagged fish released, n is the number of fish caught in the second sample, and N is the total number of fish in the pond.

Solving for N

Now that we have the formula for p, we can substitute it into the formula for N:

N = (M / m) * (n / p)

Substituting the formula for p, we get:

N = (M / m) * (n / ((m / M) * (n / N)))

Simplifying the equation, we get:

N = (M / m) * (M / m) * (n / n)

N = (M / m)^2 * N

Subtracting N from both sides, we get:

N - N = (M / m)^2 * N - N

0 = (M / m)^2 * N - N

Adding N to both sides, we get:

N = (M / m)^2 * N