Help to understand this solution of this book

Question

I'm studying A First Course in Probability and I didn't understand the solution he gives in page 444 from the exercise 4.11 of 174, chapter 4:

The question

I'm having troubles with item (a):

Could someone give me some idea of the reasoning behind this solution? I know it should be something very simple.

Answer 1

Let's try an intuitive explanation...

FACT: A wins the first game...to get 3 winning games the longest series is the following

A-BBAA

Thus the tournament could be maximum 5 games long. For A to be the winner, he has to win "at least" 2 games on 4, otherwise the winner will be B. In numbers

$$\sum_{i=2}^{4} \binom{4}{i}p^i(1-p)^{4-i}$$

But we could do a different and equivalent brainstorming.

For A to be the winner, he has to avoid that B wins at least 3 games on 4 remaining. In numbers

$$1-\sum_{i=3}^{4} \binom{4}{i}(1-p)^i p^{4-i}$$

The two solutions lead to the same result but the latter is shorter in counts with respect to the one showed in your textbook