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X-dice

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

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 $P(B> A) = \frac{5}{9}$.