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'X-dice' printed from https://nrich.maths.org/
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 $P(B> A) =
\frac{5}{9}$.