A discussion on Baseball Primer prompted me to examine the question in more detail. A contributor suggested that home field advantage was a necessary component of a first order approximation, which lead me to wonder if the order of the games mattered.
That is, does it change the overall chances of winning a series if you get to play all the home games first? More generally, if the odds of winning each game differ, does the order that you compete at those odds matter?
The answer, is no, the order doesn't matter.
I don't find this to be a particularly intuitive result, though it does fall of one of the earlier expressions of the chances of winning.
But consider swapping two games, A and B. If you look at each term of the sum that expresses the possibility of winning the series, you find three types of terms: those in which both A and B appear, and have the same "sign" (in other words, both wins or both losses); those in which both A and B appear, and have the opposite sign; and those in which only one term appears.
The magic is that when you expand the terms of the last type, you get a symmetric term, and an unsymmetric term which is reversed from one of the terms you already have.
To illustrate, consider a three game series. 1,2,3 express winning the first, second, or third game, !1,!2,!3 express losing that game. Now the odds of winning two out of three are:
12 + !123 + 1!23.
Consider swapping games one and two. The first term is symmetric in 1,2. The second and third terms are not, but together make a symmetric pair.
Now, consider swapping games 2 and 3. The third term is symmetric, so that's fine. But the second term isn't symmetric, and the first term doesn't include 3 at all.
Multiplying by one makes things more clear
12 + !123 + 1!23
= 12 (3 +!3) + !123 + 1!23
= 123 + 12!3 + !123 + 1!23
= 123 + !123 + 12!3 + 1!23
The first two terms are symmetric in 2,3, and the latter two terms make a symmetric pair.
August 29, 2004 8:49 AM
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