I am preparing for a programming exam on probability theory and I stumbled across a question I can’t solve.
Given a bag, which contains some given amount of white stones $ w$ and some given amount of black stones $ b$ , two players take turns drawing stones uniformly at random from the bag. After each player’s turn a stone, chosen uniformly at random, vanishes, and only then does the other player take their turn. If a white stone is drawn, the player, who has drawn it, instantly loses and the game ends. If the bag becomes empty, the player, who played second, wins.
What is the overall probability that the player, who played second, wins?
I assume it’s a dynamic programming question, though I can’t figure out the recursion formula. Any help would be greatly appreciated. 🙂
Example input: $ w$ = 3, $ b$ = 4, then the answer is, I believe, 0.4, which I arrived at after computing by hand all possible ways for the game to go, so not very efficient.