The first set of things to be determined was the probabilities of winning the series after each game. So, if you’re up 2-1 in games, what are your chances of winning the whole thing?
Assuming that each team has a 50% chance of winning each game (a fair simplification to make), this part is fairly simple with some statistical tact. For an example, let’s determine the chances for a team up 2-0 in the series. (For this next part, “W” means “win” and “L” means “loss” and order matters.) Up 2-0, here’s how you can win the series: WW; WLW, LWW; LLWW, WLLW, LWLW; LLLWW, LLWLW, LWLLW, WLLLW. There’s one way to win in 4 games and you have a (1/2)2 chance of doing it, so 1*1/4 = .25; there are 2 ways win in 5 games and you have a (1/2)3 chance of doing either one of them, so 2*1/8 = .25. If you complete the rest of them in this fashion, you’ll find that, with a 2-0 series lead, your chances of prevailing are .8125. The following table shows the probabilities for every single situation in a 7-game series.
This is a good start, but integrating these series win expectancies in to the games’ play-by-play account can be difficult. Again, I think this is most easily explained by example. Say, once again, you’re up 2-0. At the time of the first pitch we know your chances of winning the series are .8125. Game 3 is going to lead you to one of two situations: leading 3-0 or leading 2-1. If you lead 3-0, your chances jump to .9375, if you lead 2-1, they fall to .6875. So, to find your probability of winning the series at a moment in Game 3, you multiply your chances of winning Game 3 by .9375 and add to that your chances of losing Game 3 multiplied by .6875. This covers all the possible ways you can win the series, the set that includes winning Game 3 and the set in which you lose it. For the opening pitch of Game 3 (when the team’s chance of winning is .5), you could just trust me that your chances of winning are .8125 or you can check for yourself: .5*(.9375)+(1-.5)*(.6875) = .46875+.34375 = .8125. This can of course be done as the game situation changes and the probabilities of winning the given game and the series as a whole move up and down. Then, as with the original excitement factor, it’s just a matter of summing up the absolute value of every change that the series factor (excitement factor deluxe, from now on) undergoes.
Keep in mind, this deluxe version follows the same principles as the original excitement factor. Just as scoring lots of runs was important in having a high excitement factor, playing lots of games is important in the excitement factor deluxe because it allows for more mobility. The nature of a playoff series is also nice in that the more games the teams play the closer the series is, since blowouts end quickly. In the original excitement factor, action in the late innings was important; in the excitement factor deluxe, the later games are more important. This has a very easy mathematical explanation that agrees with the viewing experience. Let’s consider a leadoff single in the bottom of the 6th of a tie game. This raises the home team’s win expectancy from .577 to .627. In the original excitement factor, that’s worth .05. Moving to the deluxe, that same event in Game 1 would shift the probability of winning the series from .577(.6563)+(1-.577)(1-.6563) = .5241 to .627(.6563)+(1-.627)(1-.6563) = .5397, a shift of .0156. In Game 7, you don’t have to multiply the probability by all that mumbo jumbo to account for the rest of the games in the series; if you win the game you win, if not, you lose. So, in Game 7, that single would shift the probability from .577 to .627, a .05 boost. The possibility of future games dilutes the importance of the current one, so as fewer games remain, the importance grows.
The following table shows the excitement factors deluxe for the World Series from 2002 through 2006. To remind you, 2002 went the maximum 7 games, 2003 went 6, 04 and 05 were 4-game sweeps, and 06 took 5 games. As you can see, the only instance where the EFD isn’t in line with the number of games is 2005, but the White Sox and Astros played some unbelievable games in that series, though the former wound up winning all of them.
This of course isn’t a large enough sample size to draw conclusions from, but there’s something evident that we would expect to be true and I think it’s worth pointing out. The correlation between games and EFD is not linear. The 4-game 04 Series was 2.639, the 6-game 03 Series was 6.448, and the 7-game 02 Series was 9.368. As we said, the games are not of equal importance. Playing deeper into the Series not only adds more games, but it adds games that are more important/exciting.
So how about those classic ALCSs between New York and Boston, semifinal matchups that certainly resound more clearly than their ensuing finals? Let’s take a look at how exciting those really were. The EFDfor 2003 (Aaron Boone's series) was 10.597. For 2004 (the Red Sox's historic comeback) it was 6.859.
The 2003 ALCS pretty much dwarfs the one from 2004. Of course, 2004 was the one where the Red Sox made the greatest playoff series comeback in baseball history, but that wasn’t enough. One truth we found in the original excitement factor is that, within a game, one huge comeback doesn’t compare very well to multiple smaller comebacks. Evidently, this holds true for series. Amazingly, in 2004, the Red Sox won a series in which they had, at one point, a 1.9% chance of prevailing. However, that series only saw the Red Sox fall really far, then rise really high really quickly. That is not as exciting as things can get. In 2003, the ALCS went 1-0 Red Sox, 2-1 Yankees, 2-2, 3-2 Yankees, 3-3, and the Yankees ultimately won 4-3.
So, the excitement factor has met its first real obstacle, and it has conquered it.
No comments:
Post a Comment