Cards Win Game 1 - And That Means??

I broke out my Scott Rolen jersey and it paid big dividends with Rolen's second inning dinger. The Cardinals post an improbable Game One win with the rookie Anthony Reyes mastering the Tigers for 8 innings en route to a 7-2 victory. Reyes, by the way, has been a hugely talented and touted prospect so some success is not a complete surprise. Oddly, a rookie had not started a Game 1 of the World Series since 1997 and tonight both starters were rookies.

I started wondering, what does it mean to win Game 1? Not in a metaphysical sense, but in an historical or strictly statistical sense. Naturally, the answer is available on the Internet at a web site called WhoWins

Historically, when the visiting team wins Game 1, that team has gone on to win the Series 56.4% of the time (22-17). But those first game winners only go on to win Game 2 some 36% of the time. If the Cardinals do win Game 2, historically teams up 2-0 and going home have won 78.6% of the time (11-3).

Of course, not all games are equal. Tonight really figured to be the Tigers game. Reyes was a virtual throw away starter. The next three games are fairly even (at least based on the way Weaver has pitched recently).

Figuring out the 'pure' mathematical probability is somewhat more complicated (according to the folks at WhoWins):

BEST-OF-7 PROBABILITY FORMULAE
Let us say that team A is playing team B in a best-of-7 playoff series. Let us also say that team A has probability p1 of winning Game 1 vs. team B, probability p2 of winning Game 2 vs. team B, probability p3 of winning Game 3 vs. team B, probability p4 of winning Game 4 vs. team B, probability p5 of winning Game 5 vs. team B, probability p6 of winning Game 6 vs. team B, and probability p7 of winning Game 7 vs. team B in that series. Then the probability of team A winning this best-of-7 series is pWLseries, where W is the number of games team A has won in this series, and L is the number of games team A has lost in this series. Since both W and L lie between 0 and 3, inclusive, there are 16 different states for an active best-of-7 playoff series: 00, 01, 02, 03, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, and 33. Here, then, in terms of the game victory probabilities defined above (and using the notation found in the Mathematica® software package), are the corresponding series victory probabilities for team A:

Best-of-7 series with team A leading 1-game-nil: Team A series victory probability = p10series[p2_, p3_, p4_, p5_, p6_, p7_] := p2*p3*p4 + p2*p3*p5 + p2*p4*p5 + p3*p4*p5 - 3*p2*p3*p4*p5 + p2*p3*p6 + p2*p4*p6 + p3*p4*p6 - 3*p2*p3*p4*p6 + p2*p5*p6 + p3*p5*p6 - 3*p2*p3*p5*p6 + p4*p5*p6 - 3*p2*p4*p5*p6 - 3*p3*p4*p5*p6 + 6*p2*p3*p4*p5*p6 + p2*p3*p7 + p2*p4*p7 + p3*p4*p7 - 3*p2*p3*p4*p7 + p2*p5*p7 + p3*p5*p7 - 3*p2*p3*p5*p7 + p4*p5*p7 - 3*p2*p4*p5*p7 - 3*p3*p4*p5*p7 + 6*p2*p3*p4*p5*p7 + p2*p6*p7 + p3*p6*p7 - 3*p2*p3*p6*p7 + p4*p6*p7 - 3*p2*p4*p6*p7 - 3*p3*p4*p6*p7 + 6*p2*p3*p4*p6*p7 + p5*p6*p7 - 3*p2*p5*p6*p7 - 3*p3*p5*p6*p7 + 6*p2*p3*p5*p6*p7 - 3*p4*p5*p6*p7 + 6*p2*p4*p5*p6*p7 + 6*p3*p4*p5*p6*p7 - 10*p2*p3*p4*p5*p6*p7