Teams 1 and 2 are playing a match. The teams score points according to independent Poisson processes with respective rates 4 and 1. If the match ends when one of the teams has scored 5 more points than the other, find the probability that team 1 wins. I know the correct answer may use the gambler's ruin problem, with team 1 starting with 5 points, winning at 10 points, losing at 0 points, and with a probability of winning a point equal to 4/(1+4) and losing a point 1/(1+4). The probability of winning equals (1-(0.25)^5)/(1-0.25^10) which is almost 1.

Can you please solve this problem by just using the fact that the number of points for each team is a Poisson random variable and by therefore using the Probability Mass Function of a Poisson rv?