Copyright © University of Cambridge. All rights reserved.

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 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 showing how the game is played using Set A with the
interactivity (or using numbered counters in a bag). Play the game
no more than ten 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.

Use the interactivity to confirm that the experimental
probability matches closely to the theoretical probability that
students have calculated. There are opportunities here for rich
discussion about how closely we expect an experimental probability
to match the theory.

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.

Once the probabilities have been calculated, use the
interactivity again to confirm that the experimental probability is
close to the calculated one.

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

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 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.