70 1. Streaks
the 12608 completed best-of-three set matches in our data set, 4301
required three sets to complete. Of these 4301 matches, 2331 matches
were won by the player who won the second set. Thus, among the
matches that took three sets to complete, if a player won the second
set, the probability that he won the third set is .542. If the sets were
independent, we might think that this probability should be close
to .5. Since the actual probability is greater than .5, one could say
that the player who won the second set has “momentum” or is “on a
streak.” Before concluding that this is evidence of streaky behavior,
we should consider the relative abilities of the players involved in the
matches. It is possible that the reason that the winner of the second
set does so well in the third set is because he is the better player.
If we simulate a distribution of set scores, using the value of
α = .480 (obtained earlier by fitting the odds model to just the first
sets in the matches in the data), we obtain the following (the average
of 20 simulations, using the same sets of opponents as in the actual
data set):
{4676.65, 1740.05, 1187.75, 1180.4, 1746.85, 2076.3} .
In this simulated distribution, there are 5855 matches that took three
sets to complete. In these matches, there were 2935 (which is almost
exactly one-half of 5855) in which the player who won the second set
won the third. Thus, we can discount any effect due to the relative
ranks of the players.
It is also the case that this percentage does not change very much
if we vary α. For α = .3,.31,...,.5, the percentage stays between .497
and .505. Thus, we may assume that in this model, the player who
won the second set has about a 50% chance of winning the third set.
How likely is it that in as many as 2331 matches out of 4301, the
player who wins the second set wins the third set as well, given our
assumption about independence of sets? This is essentially the same
question as asking how likely a fair coin, if tossed 4301 times, will
come up heads at least 2331 times. The number of heads in a long
sequence of coin flips is approximately normal. If the coin is fair, and
Previous Page Next Page