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'Any Win for Tennis?' printed from http://nrich.maths.org/

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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.