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,
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