When Justine serves a game of tennis against Maria, Justine wins any given point 80% of the time.

• If at some point, the game is at deuce (40-40), what is the probability Justine will eventually win?
• If it's advantage Justine ("ad in" in tennis lingo--functionally equivalent to 40-30), what are her chances?
• What is her probability of winning if the advantage is to her opponent, Maria, (equivalent to 30-40)?

Scoring in tennis is, 0, 15 (one successful point), 30 (two), 40 (three). The next point is a win unless the opponent already has 40 as well, as you must win by two points. The same player serves for the entire game, so each point within one game has the same probability given. Then the serve switches for the next game in the set or match.

• What is Justine's probability of winning the game, calculated when the score is still 0-0?
• What are the probabilities from any given state of the game?

What about when Maria is serving, if she wins any given point against Justine 75% of the time?

 

At deuce, Justine has a (.8 * .8 ) .64 chance of winning the game, Maria has a (.2 * .2) .04 chance of winning and there is a (2 * .8 * .2) .32 chance a returning to deuce. Deuce again will be in the ratio of .64 : .04, so Justine will win .64/.68 times, so 94.117% of the time.

If it's advantage Justine, she has 80% chance of immediately winning, and there's a 20% chance of going to deue, which will be broken up into the same as before. So overall chances of Justine winning are 98.823%.

If it's advantage Maria, there's a 80% chance of going back to Deuce, so Justine's chances of winning are (.8 * 94.117) so 75.294%.

Posted by Claire on 2007-07-06 05:50:31