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