Two teams A and B play a game. Teams score according to a Poisson process of rates λA and λB, respectively. Whoever reaches M points first wins the match (but both teams go to the end of M points).
a) What is the probability of team A wins given that team B has reached M points after a (non-random) time?
b) What is the probability of team A wins the match?
I think that in question b, the sum of the exponential times until the M point has:
SA ~ Gamma(M, λA) and SB ~ Gamma(M, λB) distribution for team A and B respectively.
The probability that team A wins the match is P(SA < SB). Is this correct?
Any hints for the question a)? I have no idea how to solve it.
Part a says that assume Team B has reached M points at a non random time $t$. What’s the probability that A wins. I want the prob. that A reached M points before time t, which is
$$\sum_{I=M+1}^\infty \frac {e^{-\lambda_At}(\lambda_A t)^I}{I!}$$