# Probability of winning in a 20-30 Split Die Game

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?

• How do you know the answer is not correct? Perhaps in the question, Bob does not see Alice's roll (as in a similar question) Commented Sep 19, 2023 at 14:18
• The website where I encountered this question didn't accept $\frac{647}{1200}$ as an answer. Although they have reliable answers, there's a possibility that they made a mistake. I wanna confirm if I am not making any mistake. Commented Sep 19, 2023 at 14:21
• @Henry It is given that Bob can see Alice's die beforehand! Commented Sep 19, 2023 at 14:27
• Given that $B$ see's $A's$ roll, this is the same as if $B$ gets the max of the two rolls (in the sense that switching to that rule has no effect on who wins and who loses). Thus, you can just use the distribution on the max for $B$.
– lulu
Commented Sep 19, 2023 at 14:48
• Should say: I get the same answer you get.
– lulu
Commented Sep 19, 2023 at 14:53

## 1 Answer

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.

• Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. Commented Sep 25, 2023 at 19:42