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# X-dice

##### Age 16 to 18 Challenge Level:

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.