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