Suppose I have a coin for which the probability of heads is $p$. Now suppose I have a two-player game where the first player may flip this coin $m$ times and the second player may flip this coin $n$ times, and the player who gets the most heads wins.
I'm trying to determine the probability that the first player wins if $m$, $n$, and $p$ are known. Suppose $P(n, k, p)$ is the binomial probability mass function and $C(n, k, p)$ is the binomial cumulative distribution function. It would then seem that the probability of a first player win would be $\sum^m_{x=1}P(m,x,p)C(n,x-1,p)$ and the probability of a draw would be $\sum^k_{x=0}P(m,x,p)P(n,x,p)$, where $k$ is the smaller of $m$ and $n$. Are either of these expressible without the use of summation notation?
Also, is there a general name for the distribution that ensues from this (a categorial distribution where probabilities are determined by binomial distributions)?