This problem offers an opportunity to explore and discuss two types of probability: experimental and theoretical. The simulation generates lots of experimental data quickly, freeing time to focus on predictions, analysis and justifications. Calculating the theoretical probabilities provides a motivation for using sample space diagrams or perhaps tree diagrams.

The final question in the problem offers the opportunity for exploration of a rich context where collaborative working makes it possible to tackle an otherwise unmanageable task.

You may wish to use the start of What Numbers Can We Make? as a preliminary activity to get students thinking about the effect of combining odd and even numbers.

The notes that follow are based on a lesson with a group of 14 year old students. They are in two parts: the first part for teachers who wish to use the activity for a single lesson on probability and sample space diagrams or tree diagrams, and the second part for teachers who wish to follow this up with a collaborative task that leads to interesting and
unexpected results.

Start by explaining how the game is played using Set A (perhaps using numbered counters in a bag). Play the game a few times, so that students have a feel for the game but don't have sufficient results to draw conclusions about the probabilities. Then ask them to decide whether they think the game is fair, and to do some maths to support their decision.

While students are working, circulate and observe the methods being used:

Bring the class together and choose individuals who used different methods to explain what they did to the class, recording what they did on the board. Perhaps choose those who used less sophisticated methods first. Emphasise the merits of a sample space method rather than a listing method, to prepare students for tackling examples with a large number of balls. Those who are confident with
tree diagrams may prefer to continue using them.

Now show sets B, C and D, and ask them to think on their own, without writing, about which of the four sets they would choose to play with, to maximise their chances of winning. Once they have had a short time to reflect, ask them to discuss in pairs their choice, and to justify their decisions (again, without writing). There is often disagreement
about which set offers the best chance of winning, so bring the class together to compare ideas before setting them the task of calculating the probabilities - discourage them from using inefficient listing methods.

Now write up on the board a set E, which contains four large even numbers and two large odd numbers. Make them large enough that calculations would be offputting! Ask the class to work in pairs to calculate the probability of winning with set E, and give them a short time frame in which to do this. The intention is to alert students that the numbers themselves don't matter, but the numbers
of odds and evens is the important point. Set E has the same structure as Set C, so we already know the chance of winning. Then the class can be introduced to this sort of sample space diagram where odds and evens are collected together:

........................................................

Point out that none of the sets looked at so far yields a fair game. "How could we go about finding out whether there are any sets that would give a fair game?"

One way of organising the search is to draw up a table on the board showing different combinations of odds and evens:

Those already identified as not being fair games (sets A, B, C and D) can be crossed off. Then divide the class into groups working on different combinations and ask them to report back. Students could record combinations that have been checked on the board with a tick or a cross to show whether they are fair or not. If something has two ticks or two crosses,
it could be accepted as being confirmed. When disagreements arise, ask other groups to resolve them.

There will be opportunities while the class are working to stop everyone and share students' insights that will make the job easier. For example:

"None of the combinations with zero will work because..."

"If 3 odds and 2 evens won't work, 2 odds and 3 evens won't either, because..."

"You can't have the same number of evens and odds because..."

Eventually, there will be a sea of crosses on the board and just a few combinations that work (four, if the class have gone up to 9 balls in total). Ask the class to stop and consider what the fair sets have in common. This may lead to some new conjectures about the total number of balls, so organise the class to test the conjecture out on the next obvious total.

Once there is some confirmation about the total number of balls needed for fair games, conjectures can also be made about how these should be split into odds and evens. Students can be set to work to test examples with large numbers, using the simplified sample space method above. Draw attention to how valuable it is to work collaboratively as part of a mathematical community, and how
difficult it would have been to have reached the same insights working alone.

Although it is unlikely that many students will be able to prove their conjectures algebraically on their own, this proof may be sufficiently accessible to be worth sharing with the class. There are a number of ways of using this resource:

- To be presented as an elegant way of proving the ideas the students have discovered
- As a proof presented on the board for students to recreate for themselves after it's been rubbed out
- To be printed out and distributed to students for them to make sense of, and for them to annotate so that they can talk through the proof, line by line, for someone who hadn't met it yet.
- As a 'proof sorting' exercise where the proof is cut into sections and mixed up for students to reassemble into the correct order

Key questions

How can you decide if a game is fair?

What are the most efficient methods for recording possible combinations?

How can we make this difficult task (of finding a fair game) more manageable?

The problem In a Box offers another context for exploring exactly the same underlying mathematical structure, and could be used as a follow-up problem a few weeks after working on this one.

The first parts of this problem should be accessible to most students, and can be used for focussing on the benefits of using sample space diagrams instead of listing combinations.