TITLE: Probability greatest roll of z dice exceeding greatest roll of h dice?
QUESTION [7 upvotes]: Let's assume there are two players in a dice game; zombie and hero.
The Zombie rolls z fair 6-sided dice.
The Hero rolles h fair 6-sided dice.
If the heroes greatest dice roll is larger than the zombie's greatest dice roll, the hero wins.  Otherwise, the zombie wins.
How can I calculate the probability of the hero winning as a function of z and h (without just enumerating the answers).
Example 1:
z = 1,
h = 2
Zombie rolls (4), hero rolls (1,5).  Hero has a higher dice roll and wins.
Example 2:
z = 2,
h = 2
Zombie rolls (4,4), hero rolls (1,4).  Hero does not have a higher dice roll and loses.
Related Question: Given a die, what is the probability that the second roll of a die will be less than the first roll?

REPLY [5 votes]: Let $Z$ denote the greatest dice roll of Zombie and $H$ the greatest dice roll of Hero. Assume Zombie's rolls and Hero's rolls are independent
Then $\mathrm P(Z\leqslant n)=(n/6)^z$ for every $1\leqslant n\leqslant 6$ hence $\mathrm P(Z= n)=(n/6)^z-((n-1)/6)^z$. Likewise, $\mathrm P(H\leqslant n)=(n/6)^h$ hence $\mathrm P(H\gt n)=1-(n/6)^h$ for every $1\leqslant n\leqslant 6$. This yields
$$
\mathrm P(H\gt Z)=\sum_{n=1}^6\mathrm P(Z=n)\mathrm P(H\gt n)=\sum_{n=1}^6((n/6)^z-((n-1)/6)^z(1-(n/6)^h),
$$
that is,
$$
\mathrm P(H\gt Z)=1-\frac1{6^{z+h}}\sum_{n=1}^6(n^z-(n-1)^z)n^h.
$$