X-Dice
Explore these X-dice with numbers other than 1 to 6 on their faces. Which one is best?
Problem
A new type of 6-sided die, called an X-die, is proposed where instead of the faces being numbered 1 to 6 as usual, the faces are numbered with positive whole numbers such that their sum is 21. In this problem we will say that a die $A$ is worse than a die $B$ if and only if $P(A< B) > P(B< A)$ for a single throw. Conversely, a die $A$ is better than a die $B$ if and only if $P(A< B) < P(B< A)$ for a single throw.
Can you create an X-die which is worse than an ordinary 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
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
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
Key questions
Possible extension
Possible support