### 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.