### Capture and Recapture

How do scientists or mathematicians estimate the size of a population of wild animals?

### How Confident Are You?

What proportion of people make 90% confident guesses which actually contain the correct answer?

# Prize Giving

##### Age 14 to 16Challenge Level
Prize Giving

Ms Dickens' class has submitted business plans in a national enterprise competition, and to their great delight have been awarded two of the prizes.
However, 8 students - 2 boys and 6 girls - were involved in putting together their winning entries, so who should get the prizes?

Alesha and Jack felt they'd done the most work, so they ought to be chosen.

After considerable discussion the group came up with three ways of choosing who will get the prizes.

Which would you go for if you were Alesha or Jack?

Which do you think would be most fair if actually they hadn't done any more than anyone else?

Which do you think is the most representative of the whole group?

### Why do this problem?

This problem provides a context for exam questions which start 'A bag contains 6 red balls and 2 blue ones ...'

By focusing purely on the technicalities of probability calculations, questions like this obscure the fact that sampling methods are very important.  How a sample is chosen can skew results quite dramatically.

This problem is designed to provide a scenario where sampling with and without replacement can be investigated, and the merits of each discussed alongside the merits of choosing one representative for the girls and one for the boys.

### Possible approach

Discuss with the class the implications of Lucy's method.

What is the probability that a boy is chosen?
What is the probability that a girl is chosen?
Is this fair?  Is it representative?

Then put students into groups of 3 or 4, and have each group collect the equipment they need.

Take the students through the scenario for Ingrid's method - sampling with replacement.
Discuss how the experiment relates to what Ingrid has suggested.

Then get everyone to try it 16 times, putting their results onto a 2-way table and a tree diagram.  This worksheet can be used for recording data.
Results from the class should also be collated to see what happens when there is more data available.
What are the implications of this method?

Then introduce Paolo's method - sampling without replacement.
Discuss how the model relates to what Paolo has suggested.

Again, students should complete 16 trials in their groups, then put their data onto a 2-way table and tree diagram.
Collating results again around the class will provide a good amount of data for discussion.
What are the implications of this method?

Discussion should then focus around the advantages and disadvantages of each method, compared with simply choosing one girl and one boy (Lucy's method).

Students should be encouraged to stay with the experimental results until they have all been fully compared, and opinions about the three methods discussed.
Which is best if you are Alesha or Jack?

Students can then analyse the expected results for each method to see if what appears to be the case experimentally can be substantiated.
2-way tables and tree diagrams will help with this.

The worksheet could be used to support recording and discussing the experimental data, and then comparing with the expected results.

### Key questions

What is different about the three sampling methods?

This is an opportunity to:

• discuss the merits of stratified sampling (separately representing the girls and boys)
• make explicit the difference between independent and dependent events - when people are sampled with replacement, the two choices are independent, when they are sampled without replacement, the two choices are not independent

For each method, what is the probability that a girl is picked for the first prize?  How about a boy?

For each method, what is the probability that a girl is picked for the second prize?  How about a boy?

Are these the same or different for each method?

Which is the best option for Alesha?  How about Jack?

### Possible extension

If the first prize goes to a girl, what is the probability for each method that the second goes to a boy?
If the first prize goes to a girl, what is the probability for each method that the second goes to a girl (whether the same girl or a different one)?

How does this compare with the situation for picking a boy?

Further questions are provided on the worksheet.

Answering the extension questions does not require a knowledge of Bayes' Theorem - the answers can be found on the probability tree in each case.

### Possible support

All students should be able to carry out the experiments, once they have understood the scenario and done one or more initial trials all together.
The worksheet is designed to structure the analysis to help those who find it difficult to calculate the probabilities from the experiment.
Students should notice that the worksheet does not require them to calculate the expected frequencies for Paolo's method - this is because they are not whole numbers.  However observing that fact is the start of realising that the situation is different from Ingrid's method, and that it makes the analysis different.