I figured out that we can do this with dp and then i came up with the states of dp as follows:
dp[i][j][flag] : probability that flag(0 for queen, 1 for dragon) wins in the current move given he/she currently has i white and j black mice in the bag.
After this i tried a few transitions but i got confused very bad that's why i'm not writing that here.
I just need to know what will be the correct transition of states using my approach. Or , if there is any other easier way to declare the states.