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X-dice

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This is a fantastic problem for introducing conditional probability: the numbers are small but non-trivial and all is wrapped up in an interesting problem solving context with an intriguing result.

Possible approach

Explain the task. It will take some time to understand the meaning of the inequalities and ensure that all understand the meaning of the symbols.
 
Students should first experiment with creating lists of 6 numbers which they want on their dice.
 
It might be that students feel certain the one die is better or worse than another. However, they will need to prove this using the conditional probability formula. This might be cumbersome, but better students might discover rigorous shortcuts. This is a good thing. 
 
You could end by asking: which X-die would you choose if you were going to compete against someone else in a game of chance if you didn't know which X-die you would pick? This will raise questions of risk: some people might prefer to go for even odds; some might choose an X-die where there is a chance of uneven odds. 

Key questions

Explain in words what it means for a die to be better than another. Does this seem like a good definition of a 'better die'?
 
What is the expectation of an X-die?
 
You could end by asking: which X-die would you choose if you were going to compete against someone else in a game of chance?
 

Possible extension

The extension included in the problem is stimulating and challenging.
You could also ask: which X-die has the largest/smallest variance? Alternatively you can move on to Dicey Dice.

Possible support

Focus on the experimental side: just invent any X-die and see if it is better than a normal die. After a few tries of this, some structure should start to emerge in students' minds. 
 
 
You can read about some of the issues which might arise when teaching probability in this article.