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.)
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