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
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