How Many Fish?

Capture - Recapture
A pond contains 100 fish. We carefully net a sample of 25 fish, mark them with a permanent red spot and then put them. back in the pond.

A quarter of the fish in the pond are now marked with a red spot.

A day later, when the fish have had time to swim around the pond, we catch a second sample of 40 fish. Since a quarter of all the fish are marked we expect about a quarter of the 40 fish to be marked. We expect to find about 10 marked fish in the second sample.

These figures are shown in Table 1:

Table 1 - 25 marked fish in a pond of 100 fish

The first catch of 25 marked fish is all the marked fish in the pond. The number 10 is in brackets because it is only an estimate of the number of marked fish in the second sample. We expect to get about this many.

Since we take two samples and some fish are caught twice, this process is called CAPTURE - RECAPTURE.

From another pond containing 200 fish we catch a sample of 40 fish and mark them with a permanent blue
spot.

A day later another sample of 50 fish is caught.

Notice that we expect:
The proportion of marked fish in the second sample to be about the same as the proportion of marked fish in the pond.

Fish In the Pond
In most ponds we do not know how many fish there are altogether. Our problem is to estimate this total.

Suppose we catch a sample of six fish from a small pond, mark them with a red spot and put them back.

There are six marked fish in the pond:

Later we catch a second sample of eight fish and find two of them are marked.

The numbers are illustrated in Figure 1 and entered in Table 2.

Figure 1 - Six marked fish in a smallpond

Table 2 - Six marked fish in a smallpond

We expect the proportion of marked fish in the second sample to be about the same as the proportion of marked fish in the pond.

Look at Figure 1 or Table 2.

One way of doing this is to say:

In the second sample the number of fish (8) is four times the number of marked fish (2).

So in the pond the number of fish should be about four times the number of marked fish (6).

We estimate that there are 4 x 6 = 24 fish in the pond.

From another pond we catch and mark a sample of five fish. Later we catch a second sample of 12 fish and find two are marked.

Estimate the number of fish in the following ponds.

You may find it helpful to fill in blank tables like Table 5.

Harder Numbers
Sometimes the number of fish caught make the arithmetic difficult. It is not as easy to see the pattern. Table 3 shows the figures for B2g again.

Table 3 - Figures from B2g

We have written N for the number of fish in the pond. There are several ways of finding N.

Using the method described in B2, we get:

In the second sample the number of fish (9) is ⁹/₆ times the number of marked fish (6).

So in the pond, the number of fish (N) is about ⁹/₆ times the number of marked fish (16).

So N is about ⁹/₆ x 16 = ^{(9 x 16)}/₆ = 24 fish.

We can use this method for any figures. Look carefully where the numbers came from. Can you see that we put:

N =	(No. of fish in 2nd sample) x (No. of marked fish in pond) Number of marked fish in second sample

We can use this formula to estimate the number of fish in the pond.

For example, suppose we catch 11 fish, mark them and put them back into the pond. In the second sample we catch eight fish and find that five of them are marked.

This gives: N = ^{(8 x 11)}/₅ = ⁸⁸/₅= 17.6 fish

Since we cannot have part of a living fish, we estimate that there are about 18 fish in the pond.

Use the formula to estimate the number of fish in the following ponds:

Suppose there are no marked fish in the second sample.

Suppose all the fish in the second sample are marked.

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