The Birdwatch: Silly Numbers

Silly Numbers

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Many, many years ago I made the mistake of reading some sort of "Math Is Fun" book, and they told me how to do this:

STL				1.000				HOU
			0.582		0.418
		0.339		0.487		0.175
	0.170		0.413		0.330		0.087
0.085		0.292		0.371		0.208		0.043
	0.147		0.332		0.289		0.104
		0.193		0.307		0.121
0.604			0.179		0.128			0.396

This is a simple path-independent probability tree, and these calculations spit out that the Cardinals have a 60.4% chance at winning the LCS. I'll explain the meaning of all the other silly numbers under the fold. Suffice to say I really think it's a lot closer to 50/50 than 60/40, but I had to come up with something more complicated than "3 down, 8 to go."

UPDATE AFTER GAME 1:

STL				1.000				HOU
			1.000
		0.582		0.418
	0.292		0.500		0.208
0.147		0.396		0.353		0.104
	0.199		0.375		0.228		0.052
		0.218		0.289		0.095
0.732			0.168		0.121			0.268

UPDATE AFTER GAME 2:
You didn't think this would be easy, did you?

STL				1.000				HOU
			1.000
				1.000
			0.502		0.498
		0.252		0.500		0.248
	0.126		0.376		0.374		0.124
		0.219		0.375		0.156
0.564			0.218		0.157			0.436

UPDATE AFTER GAME 3:

STL			1.000				HOU
		1.000
			1.000
				1.000
			0.502		0.498
		0.252		0.500		0.248
	0.147		0.396		0.209
0.377		0.231		0.166			0.623

UPDATE AFTER GAME 4:

STL		1.000				HOU
	1.000
		1.000
			1.000
				1.000
			0.502		0.498
		0.292		0.210
0.170	0.170		0.122			0.830

UPDATE AFTER GAME 5:
Don't give up, don't ever give up.

STL		1.000			HOU
	1.000
		1.000
			1.000
				1.000
			1.000
		0.582		0.418
0.339	0.339		0.243		0.661

It starts at the top (probability = 1) and moves down a row each game, the horizontal direction depending on who wins (left for the Cards, right for the Stros). The numbers represent the probabilities of moving through each cell. Based on the rather unscientific method of averaging the Cardinals' and Astros' four levels of adjusted records, I've concluded the Cardinals are a true 0.591 team and the Astros are a 0.549 team. Using log5 I've concluded the Cards have a 0.542 probability of winning each game against the Astros in a neutral setting. Further I'll assume that it's plus-or-minus four percent for the homefield team. Thus the chance of a Cardinals sweep would be:

0.582 * 0.582 * 0.502 * 0.502 = 0.085.

The chances of the series being tied at 1-1:

0.582 * 0.418 + 0.418 * 0.582 = 0.487

There's a 30.7% chance of going to a Game 7. Yadi, yadi, yadi.

The various assumptions that drive this are problematic. The Astros are probably better than a 0.549 team now, six of the seven games could be started by the CY3 of Pettitte, Clemens and Oswalt, results of individual games aren't independent of each other, etc. The numbers are rough estimates, so I'm not suggesting anyone take it that seriously. I find the progression interesting though, and of course as the games are completed the 'errors' are corrected.

Enjoy the games, even if you have to find some weird way of getting FX.

Posted by Rob at October 12, 2005 12:22 AM

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Clay Davenport's Postseason Odds Monte Carlo sims are showing a hair over a 52% chance of the Cards taking the LCS right now.

http://www.baseballprospectus.com/statistics/postseasonodds.php

Posted by: Len Cleavelin at October 12, 2005 10:58 AM

Yeah, what's interesting is that his odds changed! I don't know if he's fiddling with the inputs or it's just random variation on a million trials.

Posted by: Rob at October 12, 2005 12:34 PM

Coolstandings is a fun site with similar information.

http://www.coolstandings.com/

Posted by: Robb at October 12, 2005 01:13 PM

"Yeah, what's interesting is that his odds changed! I don't know if he's fiddling with the inputs or it's just random variation on a million trials."

I'm not a mathematician, so I don't know everything there is to know about Monte Carlo sims, so take what I say for what it's worth, which ain't that much. You don't say how much the odds have changed (and they do change from day to day), but I figure there's two explanations. First, you would expect a certain amount of random variation from run to run, but probably not a significant amount just from that. What I mean being, if you run a million trials from game one on, and get the Cards winning 52.00034% of the trials, then run a million trials of the sim again from game one, and the Cards win, say, 51.9996%, that's not a big deal, and probably random variation.

Then again, as games are won/lost, of course the odds change as you run the million trial sim from that point on, as it's easier for the teams that won to win the needed games to win the series, and harder for the teams that lose to make up the deficit. But I'm sure I'm not telling you anything you don't already know....

Posted by: Len Cleavelin at October 13, 2005 07:38 AM

I noticed the same change as well, and it came a day after the Astros had clinched. I don't remember the exact amount, but I think it was more than 1%, so I'm pretty sure he did something to his formulas for some reason. Maybe he's taking starting pitchers into account, and he adjusted it based on new info in that department.

Posted by: John at October 13, 2005 10:57 AM

"...I don't remember the exact amount, but I think it was more than 1%, so I'm pretty sure he did something to his formulas for some reason."

I'm not certain that 1-2% is outside the realm of random variation. I'd expect a significant variation to be in the area of 3% or more (but IANAM, as I said).

OTOH, I think that the inputs are adjusted for lineup issues like expected changes in starting pitching, injuries, etc. (I want to say I read Davenport as saying that once), and that would have an effect, so you may be on to something.

Posted by: Len Cleavelin at October 13, 2005 09:45 PM