Distribution for difference in scores for a game

Question

Imagine a game where the winner is the first player out of two to win 10 independent rounds. We can model the game as series of realisations of a Bernoulli random variable with parameter $p$. Lets say that player one is much better than player two, so $p=0.8$. How can I get the PMF of the difference in score between the players? There would be zero probability mass below -10 and above 10, and zero mass at 0 (ties cannot occur), with most of the probability mass on the side that corresponds to player one winning.

I initially thought this could be achieved with binomial distributions, but it seems difficult as the number of trials is not constant and depends on the current score of the game. Is there an elegant solution to this?

Answer 1

You can model it as a Markov process. There are 10x10 = 100 different states (possible scores of players). You also know the transition probabilities: e.g. from state (5,3) it's 0.8 to state (6,3), 0.2 to state (5,4) and zero to all other states. At the beginning, you are at state (0,0). End states, e.g. (10,4) are absorbing, with probability 1 of transitioning to itself. From that, you can simply multiple the vector of state distribution by the transition matrix. After number of multiplication equal to the maximum round of games (19 in your case), you get the probability distribution over end states.