54 1. Streaks

For each group in the data, the p-values are calculated by using

the null hypothesis to compute the distribution of the number of

runs and then computing the probability, using this distribution, of

observing an outcome as extreme as the actual outcome. The small

sizes of the p-values show that the null hypothesis should be rejected,

i.e. there is streakiness in championship-level horseshoes.

5.4. Tennis. The game of tennis is interesting in probability theory

because it provides an example of a nested set of Markov chains. The

reader will recall that, roughly speaking, a Markov chain is a process

in which there is a set of states and a transition matrix whose entries

give the probabilities of moving from any state to any other state in

one step. The chain can either be started in a specific state or it can

be started with a certain initial distribution among the states. We

will describe the various Markov chains that make up a tennis match

and then give some results about tennis that follow from elementary

Markov chain theory. We will then look at whether or not tennis is

streaky at the professional level.

A tennis match is divided into sets; in most cases, the first person

to win two sets is the winner of the match. (There are a few profes-

sional tournaments in which the winner is the first person to win three

sets.) The set scores in an on-going tennis match can be thought of

as labels in a Markov chain. The possible scores, from the point of

view of one player, are 0-0 (at the beginning of the match), 1-0, 0-1,

1-1, 2-0, 2-1, 1-2, and 0-2. The last four of these states are said to be

absorbing states, because once the match enters one of these states,

it never leaves the state. The other four states are called transient

states, because the match does not end in any of those states. (A

Markov chain is said to be an absorbing chain if it contains at least

one absorbing state and it is possible to go, in one or more steps, from

every state to some absorbing state. In an absorbing Markov chain,

a state is called transient if it is not an absorbing state.)