World Series Probability

I play World Series Superstars and they recently challenged us to predict how each game of the series would play out. A user asked on their forum, "What do you think the odds are that someone will actually pick the entire series correctly game for game?" I attempted to find the answer online, but the majority of the websites I found were either determining how many games the series would last or predicted the betting odds.

I took it upon myself to try to figure this out, but had to consider several items first:
1) It's a best of seven game series.
2) The winner must win four games.
3) There will either be four, five, six, or seven total games played.
4) The series ends when a team wins their fourth game.
5) Consider a team has a 50/50 chance to win a game.

I enjoy statistics, but I was not sure how to solve this scenario with a binomial distribution equation or what have you, so I typed it out by hand. If the following information is accurate and complete, then you have a 1 in 70 (1.429%) chance of selecting the correct outcome.

Feel free to look over my details and let me know if you find any mistakes. My prediction is highlighted in bold red letters. Leave your prediction in the comments.

Four game series (2 outcomes) 4-0

StL wins Tex wins

C-C-C-C R-R-R-R

Five game series (8 outcomes) 4-1

StL wins Tex wins

R-C-C-C-C C-R-R-R-R

C-R-C-C-C R-C-R-R-R

C-C-R-C-C R-R-C-R-R

C-C-C-R-C R-R-R-C-R

Six game series (20 outcomes) 4-2

StL wins Tex wins

R-R-C-C-C-C C-C-R-R-R-R

R-C-R-C-C-C C-R-C-R-R-R

R-C-C-R-C-C C-R-R-C-R-R

R-C-C-C-R-C C-R-R-R-C-R

C-R-R-C-C-C R-C-C-R-R-R

C-R-C-R-C-C R-C-R-C-R-R

C-R-C-C-R-C R-C-R-R-C-R

C-C-R-R-C-C R-R-C-C-R-R

C-C-R-C-R-C R-R-C-R-C-R

C-C-C-R-R-C R-R-R-C-C-R

Seven game series (40 outcomes) 4-3

StL wins Tex wins

R-R-R-C-C-C-C C-C-C-R-R-R-R

R-R-C-R-C-C-C C-C-R-C-R-R-R

R-R-C-C-R-C-C C-C-R-R-C-R-R

R-R-C-C-C-R-C C-C-R-R-R-C-R

R-C-R-R-C-C-C C-R-C-C-R-R-R

R-C-R-C-R-C-C C-R-C-R-C-R-R

R-C-R-C-C-R-C C-R-C-R-R-C-R

R-C-C-R-R-C-C C-R-R-C-C-R-R

R-C-C-R-C-R-C C-R-R-C-R-C-R

R-C-C-C-R-R-C C-R-R-R-C-C-R

C-R-R-R-C-C-C R-C-C-C-R-R-R

C-R-R-C-R-C-C R-C-C-R-C-R-R

C-R-R-C-C-R-C R-C-C-R-R-C-R

C-R-C-R-R-C-C R-C-R-C-C-R-R

C-R-C-R-C-R-C R-C-R-C-R-C-R

C-R-C-C-R-R-C R-C-R-R-C-C-R

C-C-R-R-R-C-C R-R-C-C-C-R-R

C-C-R-R-C-R-C R-R-C-C-R-C-R

C-C-R-C-R-R-C R-R-C-R-C-C-R

C-C-C-R-R-R-C R-R-R-C-C-C-R

Update...
I must admit my shameful inability to count as I originally stated there were 10 possible outcomes in a five game series when there are actually only 8. I corrected the error thanks to UofMWolv25 from Playfish forum. I can now support this claim as well.

Four game series:
One team must win the first four games in a row.
1/2^4 + 1/2^4 = 2/16

Five game series:
One team must win exactly 3 out of the first 4 games, and then win the fifth game.
C(4,3) 1/2^5 + C(4,3) 1/2^5 = 8/32

Six game series:
One team must win exactly 3 out of the first 5 games, and then win the sixth game.
C(5,3) 1/2^6 + C(5,3) 1/2^6 = 20/64

Seven game series:
One team must win exactly 3 out of the first 6 games, and then win the seventh game.
C(6,3) 1/2^7 + C(6,3) 1/2^7 = 40/128

If you add the numerator from each possible series (2, 8, 20, 40), then you get 70.

1 in 70 chance (1.429%)