Ballot theorem

Among words with $a$ up-steps and $b$ down-steps, $a>b$, the number whose running balance is strictly positive after every step is $\frac{a-b}{a+b}\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)$. For nonnegative running balance with $a\ge b$, the count is the ballot number $\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)-\left(\genfrac{}{}{0ex}{}{a+b}{b-1}\right)$. These are different boundary conventions; the nonnegative formula follows by reflecting the prefix through the first step below zero.