64 1. Streaks

0 50 100 150 200 250 300

Rank difference

0

0.2

0.4

0.6

0.8

1

Probability

of

winning

Figure 30. Observed winning probabilities vs. rank difference

in 2002

As a function of log(ratio), this is a line through the origin with slope

α. The parameter α is to be determined by the data. The above

relation is equivalent to the relation

log(O(r, s)) = log(odds of success) = α log(s/r) .

This is a line through the origin with slope α.

We carry out this regression for all matches in the years 2000

through 2002 in which the ratio of the ranks of the players is at

most 7.4. (This is a completely arbitrary cut-off; it is about

e2.)

There were 12608 matches in this data set. We obtain a value of

α = .480 and a correlation coeﬃcient of .363. Figure 31 shows the

observed probability of winning the match versus the ratio of the

ranks, together with the graph of the function

(ratio)α.

Now that we have a way to estimate the probability that a given

player will win the first set against a certain opponent, we can pro-

ceed in several directions. First, we can explain and test the model

of Jackson and Mosurski, which they call the odds model. Second,