Probability of Peter Winning by 6 games to 3 or 6 games to 4 in a Tennis Match

[Larrytsai]

Peter and Alex plays tennis. Peter serves through out the first game, Alex serves throughout the second game. When Peter serves, the probability that he wins is 0.8. When Alex serves first the probability that Peter wins is 0.4. A game cannot be drawn.

After 6 games Peter and Alex both have won 3 games each. They will continue playing until one of them has won 6 games. Find the probability that Peter will win by either 6 games to 3 or 6 games to 4.

...so i have broken this question into these events.

Peter wins the next 3 games in a row, or

Peter wins 3 while losing only 1 match to alex.

Describing those scenarios I have formed a sample space describing whether Peter has won or loss denoted by 'W' and 'L' respectively.

{
WWW
LWWW
WLWW
WWLW
}

[EDIT]
so I know Peter will serve starting game 4, and it will alternate so I know the probability of each case described in sample space.

WWW = 0.8 x 0.4 x 0.8
and the rest will be = 0.8 x 0.4 x 0.8 x 0.4

now from here I do not know what to do, can someone clarify if my thought process is correct, and shoot me in the right direction?Thanks

 

[PeterO]

Peter and Alex plays tennis. Peter serves through out the first game, Alex serves throughout the second game. When Peter serves, the probability that he wins is 0.8. When Alex serves first the probability that Peter wins is 0.4. A game cannot be drawn.

After 6 games Peter and Alex both have won 3 games each. They will continue playing until one of them has won 6 games. Find the probability that Peter will win by either 6 games to 3 or 6 games to 4.

...


so i have broken this question into these events.

Peter wins the next 3 games in a row, or

Peter wins 3 while losing only 1 match to alex.

Describing those scenarios I have formed a sample space describing whether Peter has won or loss denoted by 'W' and 'L' respectively.

{
WWW
LWWW
WLWW
WWLW
}

[EDIT]
so I know Peter will serve starting game 4, and it will alternate so I know the probability of each case described in sample space.

WWW = 0.8 x 0.4 x 0.8
and the rest will be = 0.8 x 0.4 x 0.8 x 0.4

now from here I do not know what to do, can someone clarify if my thought process is correct, and shoot me in the right direction?


Thanks

Not too sure about you statement I have highlited red.

WLWW for a start will be 0.8 x 0.6 x 0.8 x 0.4 which is more than your answer, and there are the other two options to come. ?