Prize giving
Problem
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?
Lucy's suggestion
Put the boys' names on 2 cards, mix them up, and choose one. Then do it again with 6 cards for the girls.
Ingrid's suggestion
Give everyone a number from 1 to 8. Then use the random number generator on a calculator to choose two people.
Paolo's suggestion
Put everyone's names on slips of paper, and fold them over so the names can't be seen. Then mix them up, and pick two out.
Modelling the suggestions
Start by doing some experiments to help you get a feel for the different suggestions.
You should carry them out 16 times each.
Choose four colours of counters or multi-link cubes. Take 5 in one colour for the girls (except Alesha) and 1 in a different colour for Alesha. Then take 1 each of the other colours for Jack and Paolo, the two boys. (Remember to make a note of who each colour represents!).
For Lucy's suggestion, divide the counters into girls (including Alesha) and boys (both Jack and Paolo).
Without looking at the counters, pick one boy and then one girl.
Record the results, noting if either Alesha or Jack are chosen.
Repeat 16 times.
For Ingrid's suggestion, put all the counters together.
Without looking at the counters, pick one out and record what it is.
Put it back with the other counters, and then pick another, recording what it is.
Record the results, noting if either Alesha or Jack are chosen.
Repeat 16 times.
For Paolo's suggestion, put all the counters together.
Without looking at the counters, pick one out and record what it is.
This time, do not replace it with the other counters.
Pick another counter, recording what they both are.
Record the results, noting if either Alesha or Jack are chosen.
Repeat 16 times.
Which method worked out best for Alesha? How about Jack?
Without doing any further analysis yet, which do you think is most fair?
Which do you think is most representative of the group?
What should we expect to happen on average for each suggestion? This worksheet will help you to keep track of your data and to compare it with the expected results.
For each suggested method, what is the probability that Alesha gets a prize?
How about Jack?
For each suggested method, what is the probability that any one girl and any one boy get a prize?
For each suggested method, what is the probability that the same person will get both prizes?
Which method of the three would you choose? Why?
Are there any you definitely would not choose? Why not?
Can you think of a better method of choosing who should have the prizes?
How is your method better?
Does it have any disadvantages?
Teachers' Resources
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.