Capture and recapture
How do scientists or mathematicians estimate the size of a population of wild animals?
Age
11 to 16
Challenge level
Problem
Image
During 2012, the British government has been considering culling badgers in parts of the UK, because of fears that they help spread TB among cattle. In October, the proposed cull was delayed until summer 2013. One reason given for the delay was the unexpectedly high numbers of badgers.
But how do scientists estimate the number of wild animals in a given area?
One way is to use the method of capture-recapture. Suppose scientists trap and electronically tag a number of badgers on one occasion, then release them back into the wild. They then set up detectors with cameras in several locations around the forest. Some time later, the scientists return to collect the cameras. They look at all the images, recording how many badgers there were, and how many of them were tagged.
In this simulation we're going to use a bag of coloured counters (or multi-link cubes, or similar small objects with different colours). Choose one colour - this is going to represent the tagged badgers. The bag represents the area being sampled.
- Count how many counters there are altogether of your chosen colour - this corresponds to tagging badgers on the first occasion. Record this number.
- Shake the counters up in the bag.
- Scoop out a good handful of counters and count how many of your particular colour there are in your handful, and how many counters there are altogether. Record both values. This corresponds to counting how many badgers were in the pictures and how many of them were tagged.
Can you see how to estimate the size of the population from them?
Repeat the experiment a few more times.
What estimates do you get for the total population of counters in the tub?
You could calculate the average of your estimates and their spread.
Now count how many counters there are altogether in the tub.
How good were your estimates?
How well did the average you calculated describe the population of counters?
If you take account of the spread of your estimates, is the actual number within that range?
What problems do you see in using a method like this to estimate the population of a wild species in a particular area?
Can you think of any ways to refine the method to make it more accurate, or any other situations in which it might be useful.
This was essentially the method used by the Allies to gather intelligence on how many tanks the Germans were manufacturing during WWII.
Johnnie Ball's video about estimating the number of black cabs in London.
Teachers' Resources
Why do this problem?
This problem provides an experimental context to introduce students to one method of estimating populations, where direct counting is not feasible. The method relies on proportional reasoning, and emphasises the value of using an average of several samples to provide a more reliable estimate.
Possible approach
Provide each group of students with a bag (not transparent) containing a reasonably large number of coloured counters (or small multi-link cubes or similar). The bag should contain a number of red (or some other particular colour) counters or cubes - the colour of the rest doesn't matter.
The bag represents the area being sampled.
The red counters/cubes represent the badgers which have been tagged.
The rest of the counters/cubes represent the rest of the population in the given area.
Students start by counting the number of red counters/cubes.
They then put all the counters/cubes back in the bag, and shake them up.
They remove a handful and count how many red ones there are in their sample.
They then count how many counters/cubes there are altogether in their sample.
From the three recorded figures - the number of tagged badgers, the number of tagged badgers in the sample, and the total number of badgers in the sample, they should be able to estimate how many badgers there are altogether in the area:
We assume that the proportion of tagged badgers in the sample is equal to the proportion of tagged badgers in the area.
Key questions
If you know the proportion of tagged badgers in the sample, how does that help you to estimate how many badgers there are altogether in the given area?
Why might one sample only not provide a good enough estimate?
If you take several samples, how can you combine the results to give a better estimate?
Possible extension
Students should criticise the model and its assumptions.
Possible support
Proportional reasoning can be conceptually difficult for students.
Rather than using a formula, which simply obscures what is going on, help students who find this difficult with a series of questions:
If half the badgers in the sample are tagged, what proportion of the whole population of badgers would we expect to be tagged?
How many badgers were tagged?
So how big would the population be?
How many badgers were tagged?
So how big would the population be?
Then vary the proportion, perhaps to a third, or a quarter, to establish the principle that whatever proportion are tagged in the sample, we would expect the same proportion to be tagged in the population. We know the proportion in the sample, so we use that to estimate how many badgers there are in the population.