Problem

Two teams AA and BB play a soccer match. The number of goals scored by Team AA is modeled by a Poisson process N1(t)N1(t) with rate 1=0.021=0.02 goals per minute, and the number of goals scored by Team BB is modeled by a Poisson process N2(t)N2(t) with rate 2=0.032=0.03 goals per minute. The two processes are assumed to be independent. Let N(t)N(t) be the total number of goals in the game up to and including time tt. The game lasts for 9090 minutes.

1. Find the probability that no goals are scored, i.e., the game ends with a 00-00 draw.
2. Find the probability that at least two goals are scored in the game.
3. Find the probability of the final score beingTeamA:1,TeamB:2TeamA:1,TeamB:2
4. Find the probability that they draw.