66 1. Streaks
Note that if we take k = 1, then we have a model in which the sets
are assumed to be independent.
The above equation can be used, along with the data, to estimate
α and k. The procedure we now describe results in the maximum
likelihood estimates for α and k. In a nutshell, a maximum likelihood
estimate for a set of parameters for a family of distributions is the set
of values for those parameters that leads to the highest probability,
among all distributions in the family, that the actual data set would
occur. As an example of this idea, suppose that we flip a coin 20
times and observe 12 heads. We wish to find the maximum likelihood
estimate for p, the probability of a head on a single toss. We certainly
hope that this estimate is 12/20. Let us see if this is the case. For
each p between 0 and 1, we compute the probability that if a coin has
probability p of coming up heads, it will come up heads 12 times in
20 tosses. This probability is
20
12
p12(1

p)8
.
We wish to maximize this expression over all p. If we denote this
expression by f(p), then we have
f (p) =
20
12
12p11(1

p)8

8p12(1

p)7
.
Setting this equal to 0 and solving for p, we obtain
p =
12
20
,
as we hoped. (It is easy to check that this value of p corresponds to a
maximum value of f(p).) To reiterate, this means that the probability
that we would actually obtain 12 heads in 20 tosses is largest if p =
12/20.
In the case of set scores in tennis, we treat each match as an
independent event and use the relationship given above among the
odds that the better player wins a given set, α, and k, to compute
the probability that we would obtain the actual data set. Since the
matches are assumed to be independent, this probability is the prod-
uct of the probability that each match in the data set occurs, given
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