KURSCHAK 2015 Problem 1

In fencing, you win a round if you are the first to reach $15$ points. Suppose that when $A$ plays against $B$ , at any point during the round, $A$ scores the next point with probability $p$ and $B$ scores the next point with probability $q=1-p$ . (However, they never can both score a point at the same time.)

Suppose that in this round, $A$ already has $14-k$ points, and $B$ has $14-\ell$ (where $0\le k,\ell\le 14$ ). By how much will the probability that $A$ wins the round increase if $A$ scores the next point?