Prob question

[lynt.br]

I figure I'm missing something extremely obvious in this question but I cannot get the answer the book says!

Chapter 18 Probability Revision Extended Response question 3c part i in the Methods (Non CAS) Essentials book.

Lleyton Hewitt and Roger Federer are playing the final of a tennis tournament. The winner of the match will be the first player to win two sets. The results of each set follows a Markov sequence, as described below

Pr(L₁) = 0.45
Pr(R₁) = 0.55
Pr(L_i+1|L_i) = 0.6
Pr(R_i+1|L_i) = 0.4
Pr(L_i+1|R_i) = 0.45
Pr(R_i+1|R_i) = 0.55

What is the probability that Lleyton Hewitt will win the match given that he wins the first set?

By my logic, this leaves two possible outcomes:
Leyton wins the next game.
Pr(L_i+1|L_i)

Roger wins the next game, Lleyton wins the one after that.
Pr(R_i+1|L_i) x Pr(L_i+1|R_i)

The combined probability of these two possible outcomes would then be
Pr(L_i+1|L_i) + [Pr(R_i+1|L_i) x Pr(L_i+1|R_i)]
= 0.6 + (0.4 x 0.45)
= 0.78

The answer in the back of the book says 0.6

What am I missing here?

[/0]

Yeah that seems right. Maybe it's because the data is old, I'm pretty sure he'd have a lot lower chance of winning now.