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Same or Different?

Anna and Becky were playing a game. They put one purple cube and two yellow cubes into a bag.

one purple and two yellow cubes

First Anna picked a cube out of the bag without looking, then Becky picked one out.
If the two cubes the girls picked out were the same colour, Anna won the game. If they picked out two differently coloured cubes, then Becky was the the winner.

Is this a fair game?
Explain your answer.

Thank you to Doug Williams from the Mathematics Task Centre Project for this problem. If you'd like to take this idea a bit further, look at In a Box.


Why do this problem?

This problem provides a context in which pupils can explore the likelihood of events and therefore gain experience in using appropriate vocabulary.

Possible approach

You could set this problem up by acting it out with two children. Put three cubes, two of one colour and one of a different colour, in a bag and explain how the game works. Play a few times and ask the whole group to consider whether they think it is a fair game or not. Their answers may well depend on the outcomes of the games they have seen, for example someone may suggest that it is fair because each child happened to win twice each. In other circumstances, someone may suggest it isn't fair because one child won more games than the other.

Ask learners what they would expect to happen if you played many, many more games. Give them time to think about this in pairs and have cubes available for them to work with should they want them. As the children discuss the problem, listen out for those who are looking at all the possibilities and analysing each in turn.

In the plenary, invite some pairs to share their methods and explanations for their solution. It would be helpful for them to write on the whiteboard as they talk so that the processes are recorded. Some may justify their answer in terms of numbers of possible ways that Anna/Becky can win. Some may have given probabilities in terms of fractions for each girl winning, but this is not necessary for a full and convincing solution.

Key questions

What colour cube could Anna pick from the bag? What are Becky's options then?
What are all the different ways that two cubes could be taken from the bag?

Possible extension

Children could try In a Box which takes these ideas a bit further.

Possible support

Encouraging children to list all the different ways to pick out two cubes will help them access this problem.