Higher GCSE

# The Capture-Recapture Method – how to estimate the number of fish in a lake

In the previous blog, I explained the Difference of Two Squares, as this mentioned in the Advance Information. Another topic that is mentioned specifically in the Edexcel Advance Information for the Summer 2022 GCSE exams is the Capture-Recapture Method. This will be very easy to prepare for as the questions are usually very similar and follow the same pattern. In this blog I will explain the concept and work through some example questions.

Please note: this topic will only appear in the 2022 Edexcel Higher Tier GCSE – so if you are taking the Foundation exam or the AQA exam, this blog will be of no use to you!

## What is the Capture-Recapture Method?

In broad terms, the Capture-Recapture Method is a very clever way of estimating the population or number of items in a region. The best way to explain what the Capture-Recapture Method is, is by way of an example.

So let’s say you are standing at a lake and you want to estimate how many fish there are in there. This is a nigh-on impossible task, but your best bet is to use this method.

First of all, I would take a sample of fish from the lake. Let’s say 50 of them. I will mark these fish somehow, for example by clipping a fin or tagging them. It doesn’t really matter how, all we need to do is make it so these fish are identifiable later. Then I return those 50 fish to the lake.

The next day, I will go back to the lake and take another sample of fish, let’s say for example 200 fish. Some of these fish will be marked from yesterday. Let’s say for simplicity that 20 of the fish I catch are marked. I know that yesterday I marked 50 fish. Today, 20 out of 200 (or 10%) of the fish I caught were marked. So based on my sample it would be fair to assume that 10% of the fish in the lake are marked. So, if x is the total number of fish in the lake,

10% of x = 50

So…

x/10 = 50

Now we need to solve the equation to find x:

x = 50 x 10 = 500

So I estimate that there are 500 fish in the lake.

## What assumptions does the method make?

Obviously this method is far from perfect and shouldn’t be relied on as the gospel truth (it is an estimate after all!). There are a few assumptions that we have made in making this calculation (sometimes you will be asked to state an assumption in an exam):

• The fish swim around randomly and do not stay in the same area of the lake
• None of the marked fish have died between the first and second samples
• No new fish have been born between the first and second samples

## An exam-style question

Chris wants to estimate the number of beetles in a garden.
He catches and marks 25 beetles, then returns them to the garden.
The next day Chris catches 40 beetles and notices that 9 of them are marked.

Work out an estimate for the total number of beetles in the garden.

Chris went back and noticed that 9/40 of the beetles are marked.

Chris marked 25 beetles, so we can assume that, if x is the total number of beetles:

So, we can form this equation and solve it to find x:

So rounding to the nearest whole number, we estimate that there are 111 beetles in the garden.

These questions are very formulaic so with a bit of practice you should be in a great position to pick up the marks when this question comes up in the Summer 2022 Edexcel higher tier exam (my guess is that it will come up in one of the two calculator exams, but in theory it could also come up in the non-calculator exam if the numbers are nice). There aren’t many resources for this topic online, but your best bet is with Corbett Maths – here is his textbook exercise.