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?

Numerical extension

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?

Explore.

NOTES AND BACKGROUND

In the 2010 Wimbledon Championships, Isner and Mahut played the longest match in tennis history: the match went on for three days and finished with a score of 70-68! (You can read about it in the Plus article here) After the match, Isner said that a match like this will never happen again.

I wonder if Isner was correct in this statement. The famous Cambridge mathematician Tim Gowers thought about this question on his blog.

In the 2010 Wimbledon Championships, Isner and Mahut played the longest match in tennis history: the match went on for three days and finished with a score of 70-68! (You can read about it in the Plus article here) After the match, Isner said that a match like this will never happen again.

I wonder if Isner was correct in this statement. The famous Cambridge mathematician Tim Gowers thought about this question on his blog.