Suppose a game is played in which Player 1 must gain two points to win and Player 2 must gain five points to win. Both players start with zero points. In any round, Player 1 has a $1/3$ chance of getting a point while Player 2 has a $2/3$ chance of getting a point. How can the probabilities at each state of this game be represented mathematically? Is this something that Markov chains would be used for?
2026-04-02 03:20:55.1775100055
Determining the Likelihoods of Different Game States
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Since there only two players, one of whom gets a point each turn, you can represent the entire game from the perspective of only one of the players. Lets do player 1:
Player 1 wins if he gets two wins before five losses, and he has a probability of winning of 1/3. Therefore, the longest game will last six turns - whoever met their quota in these six turns wins.
We can model the score of Player 1 as a binomial distribution: $S_1 \sim Bin(6,1/3)$, then the probability of Player 1 winning is $P(S_1 \geq 2)$ and the probability of player 2 winning is just the complementary probability.