Alice and Bob have fair 30−sided and 20−sided dice, respectively. The goal for each player is to have the largest value on their die. Alice and Bob both roll their dice. However, Bob has the option to re-roll his die in the event that he is unhappy with the outcome. He can see Alice's die beforehand. Bob then keeps the value of the new die roll. In the event of a tie, Bob is the winner. Assuming optimal play by Bob, find the probability Alice is the winner.
If Alice gets a number between 21-30, they win no matter what Bob gets. Otherwise, if Alice gets a number x (<21) on the dice, then the probability of them winning should be $\frac{(x-1)^2}{400}$, that is Bob gets a number less than x in both the throws. Therefore the probability of Alice's win is $\frac{1}{3} + \sum_{x=1}^{20}\frac{1}{30} \cdot \frac{(x-1)^2}{400} = \frac{647}{1200}$. But this answer isn't correct, can someone help me point out my mistake?
The website (if it is quantguide.io) has the question written wrong. It should say that Bob cannot see Alice's roll before re-rolling. If you solve for the probability that way you will get "the right answer" according to the website.