5. Data from Various Sports 59

Theorem 1. Let bij be the probability that an absorbing chain will

be absorbed in the j’th absorbing state if it starts in the i’th transient

state. Let B be the matrix with entries bij. Then B is an t-by-r

matrix, and

B = NR ,

where N is the fundamental matrix and R is as in the canonical form.

As an example of how this theorem is used, suppose that the

first player has probability p = .6 of winning a given set against his

opponent. Then

Q =

⎛

⎜

⎜0

⎝0

0 .6 .4 0

0 0 .4

0 0 .6

0 0 0 0

⎞

⎟

⎟

⎠

,

so one can calculate that

N =

⎛

⎜

⎜0

⎝0

1 .6 .4 .48

1 0 .4

0 1 .6

0 0 0 1

⎞

⎟

⎟

⎠

.

Thus, the matrix B = NR is given by

B =

⎛

⎜

⎜

⎜

⎝

2-0 2-1 1-2 0-2

0-0 .36 .288 .192 .16

1-0 .6 .24 .16 0

0-1 0 .36 .24 .4

1-1 0 .6 .4 0

⎞

⎟

⎟

⎟

⎠

.

The first row of this matrix is of particular interest, since it con-

tains the probabilities of ending in each of the absorbing states, if the

chain starts in the state 0-0 (i.e. the match starts with no score). We

see that with the given value of p, the probability that the first player

wins both of the first two sets is .36 and the probability that he wins

the match is .36 + .288 = .648.

There are 43 states in the Markov chain representing a set of

tennis. There are four absorbing states, with two corresponding to a

win by the first player and two to a win by the second player. The