# Computing the probability of winning the grand prize in a hypothetical game

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?

• A tree diagram showing possible outcomes (after each round, conditional on outcome in previous round) and their probabilities could be helpful for your calculations. – AlexK May 20 at 7:28
• See this question. – Bayequentist May 20 at 8:41
• Some useful search terms for elementary solution methods include "recursion" and "dynamic programming." On our site you can find many examples of these methods (although they are not always explicitly labeled as such) by searching for game probability – whuber May 20 at 12:38