### Tennis Outcomes Related to Probability of Winning Individual Points

Suppose in every point of a match you had a 50% probability of winning every point.  If you played thousands of such matches you expect to win as many matches as you lose.  Furthermore, you expect that much more of the time the scores will be close, like 6-4 rather than 6-0.  What are the scores like and how often do you win for other probabilities of winning each point?  The following graphs show the results for relevant probabilities.  For example, if your probability were 70%, you virtually never lose a set, and 60% of the time you win a set at 6-0.

These results  were generated by a computer program that used a random number generator to “play” points according to the different probabilities and which then kept score, using a 7 point tie-breaker at 6-6.  For each probability, 1,000 matches were played.  An interesting result is that even with a very modest improvement in one’s probability of winning a point, one’s probability of winning the match is very dramatically improved.  For example, if one’s probability of winning a point improves from 50% to 55%, the probability of winning a set improves from 50% to 84%.