I'm a beginner to probability so please be a bit gentle on me. I have trouble finding the correct approach / search terms to find related problems.
I face the following problem. In a hypothetical game the player gets a number of tokens with which he or she can play the game. Every round of the game costs the player 1 token to play. In every round the player either wins nothing (and thus effectively loses 1 token), wins 2 more tokens to continue playing, or wins the grand prize after which the player will stop playing the game. Now let's say that the game itself is a black box, but we can describe the outcomes of the game probabilistically with some chance to win nothing, some chance to win 2 more tokens, and some chance to win the grand prize. Now what I'm interested in is, given these win probabilities and an initial state of the player (e.g. the player starts with 2 tokens), to compute the probability that the player will win the grand prize. How do I go about this?