60 1. Streaks

0 0.2 0.4 0.6 0.8

Point probability

0.2

0.4

0.6

0.8

1

Set

probability

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.