GCSE Maths: Capture-Recapture Method Exam Question Walkthrough

If you're studying GCSE maths, you've likely encountered questions about estimating populations using the capture-recapture method. This statistical technique appears in exams from AQA, Edexcel, and OCR, and it's a favourite topic for examiners. Let's walk through a typical question step by step.

Understanding the Capture-Recapture Method

The capture-recapture method (also called mark-recapture) is used to estimate the size of a population when counting every individual is impractical. Wildlife biologists use it to estimate animal populations, and in maths exams, you'll often see it applied to beads in a bag, fish in a lake, or similar scenarios.

The basic principle is simple:

Capture a sample, mark them, and release them back into the population

Later, capture another sample

Count how many marked individuals appear in the second sample

1. Use proportion to estimate the total population

The Exam Question

Let's look at our specific question:

> Bhavna has a bag containing a large number of beads.
> She wants to find an estimate for the number of beads in the bag.
> Bhavna takes a sample of 30 beads from the bag.
> She marks each bead with a black cross.
> She then puts the beads back in the bag.
> Bhavna shakes the bag.
> She now takes another sample of 30 beads from the bag.
> 4 of these beads have been marked with a black cross.
> (a) Work out an estimate for the total number of beads in the bag.
> (b) State any assumptions you have made in your answer to part (a).

Step-by-Step Solution

Part (a): Calculating the Estimate

Let's break this down:

First sample (capture): 30 beads are marked and returned to the bag

1. Second sample (recapture): 30 beads are taken, and 4 of them are marked

We can set up a proportion:

flowchart TD
    A["Marked beads in population: 30"] --> B["Total population: N"]
    C["Marked beads in sample: 4"] --> D["Sample size: 30"]
    B -- proportional to --> D
    A -- proportional to --> C

The proportion of marked beads in the second sample should equal the proportion of marked beads in the whole population:

^{Number marked in second sample}⁄_{Size of second sample} = ^{Number marked in whole population}⁄_{Total population}

Substituting our numbers:

⁴⁄₃₀ = ³⁰⁄_N

Where N is the total number of beads we're trying to estimate.

Now we solve for N:

⁴⁄₃₀ = ³⁰⁄_N

Cross-multiply:

4 × N = 30 × 30

4N = 900

N = ⁹⁰⁰⁄₄ = 225

So our estimate for the total number of beads is 225.

Alternative Approach: The Formula

Many students find it helpful to remember the formula:

Estimated population = ^{(Number in first sample) × (Number in second sample)}⁄_{Number marked in second sample}

N = ^{30 × 30}⁄₄ = ⁹⁰⁰⁄₄ = 225

Both methods give the same result.

Part (b): Stating Assumptions

For part (b), we need to state the assumptions that make our calculation valid. These are crucial for getting full marks:

1. The marked beads mix evenly with the unmarked beads - This is why Bhavna shakes the bag. If marked beads clump together, our sample won't be representative.
2. The marking doesn't affect the beads' likelihood of being selected - The black cross shouldn't make beads more or less likely to be chosen in the second sample.
3. The population remains constant between samples - No beads are added or removed between the first and second samples.
4. The marking doesn't wear off or become invisible - All marked beads should still be identifiable in the second sample.
5. The sampling is random - Every bead has an equal chance of being selected in both samples.

In exam answers, you should mention at least two or three of these assumptions. The most important ones are the even mixing and constant population.

Common Mistakes to Avoid

1. Reversing the proportion - Don't write ³⁰⁄₄ = ^N⁄₃₀. Always check that marked/total in sample equals marked/total in population.
2. Forgetting to state assumptions - Part (b) is often worth 1-2 marks, so don't skip it!
3. Rounding too early - Keep calculations exact until the final answer. Here we got a whole number, but sometimes you might get decimals that need appropriate rounding.
4. Confusing samples - Remember which numbers go where: first sample size, second sample size, and marked in second sample.

Practice Makes Perfect

Try this similar question:

> A biologist wants to estimate the number of fish in a lake. She catches 50 fish, tags them, and releases them. A week later, she catches 40 fish and finds 8 are tagged. Estimate the total number of fish in the lake.

⁸⁄₄₀ = ⁵⁰⁄_N

8N = 2000

N = 250

Estimated population: 250 fish

Exam Tips

Show your working clearly - Even if your final answer is wrong, you can get method marks

Label your answer - Write "Estimated total = 225 beads" or similar

Use the formula if it helps - But understand where it comes from

• Practice different contexts - Capture-recapture appears with beads, fish, insects, and even in social surveys

Summary

The capture-recapture method is a straightforward but important statistical technique in GCSE maths. Remember:

Set up the proportion correctly: marked in sample/sample size = marked in population/total population

Solve the resulting equation

1. State the necessary assumptions for the method to be valid
2. With practice, you'll be able to tackle these questions confidently in your exam. Good luck with your revision!