60 1. Streaks
0 0.2 0.4 0.6 0.8
Figure 27. Set-winning probabilities for players of equal abilities
reason that we need two winning states for each player is that we need
to keep track of who serves first in the subsequent set (if there is one).
The transition probabilities are found by using the graph shown in
Figure 25; for example, if the first player is serving, the game score is
2-1, and he has a probability of .6 of winning a given point, then the
game score will become 3-1 with probability .74.
Once again, we are interested in the probability that the player
who serves first wins the set. This time, there are two parameters,
namely the probabilities that each of the players wins a given point
when they are serving. We denote these two parameters by p1 and
p2. If we let p1 = p2, then Figure 27 shows the probability that the
first player wins the set as a function of p1.
There is nothing very remarkable about this graph. Suppose
instead that p1 = p2 + .1, i.e. the first player is somewhat better than
the second player. In this case, Figure 28 shows the probability that
the first player wins the set as a function of p1 (given that the first
player serves the first game). If, for example, p1 = .55 and p2 = .45,
then the probability that the first player wins the set is .81.