Does the following sum simplify?
$p(x,y\mid a,b,s)=\sum_{k=\max(0,a-x)}^{\min(a,y)} \binom{a}{k} \binom{b}{x-a+k}s^{2k+x-a} (1-s)^{y+a-2k}$
For context, this models the scenario where we initially have $a$ switches in the 'on' position and $b$ switches in the 'off' position, with each switch having a probability $s$ of switching states. $p(x,y\mid a,b,s)$ is the probability that, post switching, we have $x$ switches in the 'on' position and $y$ switches in the 'off' position.
Thanks.