X-Dice
Explore these X-dice with numbers other than 1 to 6 on their faces.
Which one is best?
Age
16 to 18
Challenge level
Problem
Image
Can you create an X-die which is worse than an ordinary die?
Can you create an X-die which is better than an ordinary die
using only the numbers $1$ to $6$ (you don't have to use all of the
numbers!)?
Be clear in your explanations or reasoning.
You can prove your results using algebra or explore the
problem experimentally using a spreadsheet.
Extension: Explore the
notion of a 'worst' or a 'best' X-die.
Getting Started
This problem involves conditional probability.
Consider two dice $A$ and $B$, where the largest number on $A$ is $N$. Then $P(A< B)$ is
$$
P(A< B) = \sum^N_{m=1}P(A< B|A=m)P(A=m)
$$
Of course, in this expression $P(A=m)$ is zero if the integer $m$ is not present on the die.
With problems such as these, don't be afraid to start with a period of experimentation: just choose any numbers to begin with and explore their properties. The structure of the problem will soon start to emerge.
Student Solutions
A wonderful solution to this problem was sent in by Mark from British School of Manilla, which we have included as a pdf for easy viewing
See Mark's Solution.
This solution is included as submitted and needed no editing or correcting. Well done Mark!
Steve says this about his problem:
There are plenty of worse X-die, the worst being (1, 1, 1, 1, 1, 16). (It will be 'obvious' but a little awkward to demonstrate this. Comparison with (1, 1, 1, 1, 2, 15) is a start) .
It transpires that if you use only the numbers 1 to 6 then no single X-die is better or worse than an ordinary die, with $P(A> B)=P(B> A) = \frac{5}{12}$.
However, one X-die with the numbers 1 to 6 can be better than another.
For example, A = (2, 2, 2, 3, 6, 6) and B = (1, 4, 4, 4, 4, 4).
For these we have $P(A> B) = \frac{4}{9}$ and $P(B> A) = \frac{5}{9}$.
Teachers' Resources
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.