d30 versus d20 with one blind reroll for Bob

Alice holds a fair $30$-sided die and Bob a fair $20$-sided die; each wants the higher number, and Bob wins ties. Both roll. Without seeing Alice's result, Bob may choose to roll his die once more, in which case the second result replaces the first. If Bob plays to maximise his chance of winning, what is the probability that Alice wins?

$17/30$. If Bob's final number is $b$, Alice wins with probability $1-b/30$. A fresh roll averages to $1-10.5/30=0.65$, so Bob rerolls exactly when $1-b/30>0.65$, i.e. $b\le10$. Then $P(\text{Alice})=\tfrac1{20}\sum_{b=11}^{20}(1-\tfrac b{30})+\tfrac12(0.65)=\tfrac{145}{600}+\tfrac{195}{600}=\tfrac{17}{30}$.