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A Dicey Paradox

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Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

Non-transitive Dice

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem offers a good opportunity to introduce or practise using sample space diagrams or tree diagrams. Transitivity is such a common phenomenon that most students take it for granted so they may be surprised and intrigued by the existence of non-transitive dice.

Possible approach

This printable worksheet may be useful: Non-transitive Dice.

Before the lesson, create a set of the three dice from the problem.
"We're going to play a game. I need a volunteer to choose one of these three dice. Then I'll choose one too and we'll roll them together - the winner is the person whose die shows the bigger number." Share with the class the numbers that are on each die.

When the volunteer has chosen one of the dice, choose the appropriate die from the two remaining. (Red beats blue, blue beats green and green beats red.)
Roll sufficiently many times for students to doubt whether this is a fair game!
"I seem to be lucky today, or perhaps my die is stronger than yours!"
"We're going to play the game again at the end of the lesson. I want you to explore whether you think the game is fair or not, and to work out a strategy for choosing dice that will give you the best chance of winning."
If it's not possible to make a set of dice before the lesson, you could introduce the task by asking students to work out how the game could be used to raise money at a school fundraiser, and what strategy the stallholders should use to fleece the public! 
As students work on the task in small groups, circulate and observe the methods they are using. For those who have difficulty getting started you could use prompts such as:
"What are the possible outcomes when red plays green? What about when red plays blue? What about when green plays blue?"
"How could you organise the information systematically?"
"Are there any diagrams you could draw that might help?" 
"How could you work out the probability of each of these outcomes?"

After a while, pause the class to share methods of approaching the task, and if appropriate introduce sample space diagrams or tree diagrams as a good organising structure for grouping the different possibilities. Then give them time to use the different methods to complete the task.
For those who finish early, challenge them to find other sets of non-transitive dice. This spreadsheet could be used to explore or to check.
Finally, bring the class together and invite them to challenge you to another game, using their strategy, and ask them to explain how they arrived at their conclusions.

Possible extension

Dicey Dice invites students to create different sets of non-transitive dice, using the numbers from 1-6. Again, the spreadsheet could be used to consider different possibilities.
A Dicey Paradox offers a set of four non-transitive dice for students to compare. Alternatively, students could be invited to devise their own set of four non-transitive dice.

Possible support

Tricky Track is a simpler context that can be used to introduce sample space diagrams or tree diagrams.