HMMT 2021 U31

Roger initially has 20 socks in a drawer, each of which is either white or black. He chooses a sock uniformly at random from the drawer and throws it away. He repeats this action until there are equal numbers of white and black socks remaining. Suppose that the probability he stops before all socks are gone is $p$. If the sum of all distinct possible values of $p$ over all initial combinations of socks is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.