Winner leading throughout a 100 to 80 count

An election ends with $100$ ballots for candidate $A$ and $80$ for candidate $B$. The ballots are shuffled uniformly at random and counted one at a time while a running tally is kept. What is the probability that after every ballot counted, $A$ has strictly more votes than $B$ (the empty tally at the start does not count)?

$\tfrac19$. This is Bertrand's ballot theorem: with $n>m$ votes the probability $A$ leads throughout is $\tfrac{n-m}{n+m}$, provable by conditioning on the last ballot and inducting on $n+m$ (or by the reflection principle). Here $\tfrac{20}{180}=\tfrac19$.