A mathematician tennis player said:
"In tennis you win a game if you score 4 points before your opponent scores 3 points. Or, if you both score 3 points at some stage you win if you manage to score 2 points in a row after the 3-all stage before your opponent does."
This sentence is quite a mouthful to say, so first think about what it means! If you play tennis, think about how this mathematically represents the scoring system.
Suppose that you have a fixed chance of $0.6$ of winning any given point. What is your chance of winning a game?
In reality a fixed chance of winning a point is not a good assumption. Suppose that Ahmed has a 60% chance of winning the first point if he serves, 80% chance of winning a point if he has just won a point and a 40% chance of winning a point if he has just lost a point. Suppose that Bryoni's chances are 85%, 80% and 30% respectively if she serves.
What chance would each player have of winning a service match?