The failing fallacy

Commenter Donald brings up some points about calling it now that I'm sure are echoed by every one still worried about how the next ~45 games are going to play out. We've seen teams lose 3-4 in a row and win 3-4 in a row all year long. If the Nats were to do something like the former and the Marlins like the latter, and the Nats were only up by 5, 5 games from now, surely I wouldn't call it then.

He's right. I wouldn't. Not with six head to head games left. But there's something at play here that we need to address. Just because we've seen teams win and lose in streaks like that does not make it likely. It just makes it possible. How possible is what drives "calling it". If that wasn't the case, you'd never call it before mathematical elimination.

So how possible is the scenario described above? Well both teams play 5 games in 5 days and the Marlins are 8.5 games out so technically 5 games out at the end isn't possible. But let's give the Marlins a rain out. Let's make the Nats go 1-4 and the Marlins 4-0.  That would do it. Those don't seem that crazy do they?

Well let's look at how many times these two teams have actually hit those marks in a 5/4 game stretch this season. We'll include overlapping streaks because well, they count too.

The Marlins have had 116 four game stretches (from games 1-4, 2-5... all the way through games 116-119) They have won all four games in those stretches 6 times. The Nats have had 114 five game stretches. They have lost four or more games 11 times. So if we take these things to be completely random you'd figure the chances of the Marlins having a 4-0 streak right now while the Nats were to have a 1-4 streak to be 6/116 * 11/114. That would be 66/13224 or real close to 0.5%.

That's not 5%, that's 0.5%, meaning there would be a 99.5% chance of that NOT happening.

But why does it feel like it could? Well part of the reason is that you aren't just following your team generally. You are following your team, and the teams that they are playing now, and probably playing next, and their divisional opponents, and maybe a couple other teams of interest.  All in all at any point in the season you may know how, let's say 8, different teams are doing.  Let's say they all have the same chances of winning 4 in a row as the Marlins. (probably not bc the Marlins are in the top half of the league but bear with me here).  The chances that any one of them are on a 4-0 streak at any given time is the inverse of the chances that all of them are not (the latter is easier to calculate and then we can just subtract from one). At any given point the chances of one of them being 4-0 is like a 1/3.  Let's say the chances of 0-4 are the same* and thus you have a 1/3 chance of one of the eight being on that streak at any moment too. The end result is that you have a pretty good shot at any one time, about 2/3rds, to find one of the teams you follow on a winning or losing streak of 4 games (or more)

It may not seem right but it is. If you are watching a bunch of teams, chances are someone is streaking. It's almost certain in fact if you are paying attention to half the league or more. If you are an AL fan Boston's won 5 in a row, LA has lost 4. Three other teams are on 3 game streaks. In the NL San Diego has lost 4 in a row and 4 other teams are on 3 game streaks. However the chances of any one specific team streaking in a certain direction and another specific team streaking in another direction at the same time are very low. They get even lower if they are going against what they have done all year.

This is just one part of why calling it makes sense though. Here's what simple logic tells us. The Nats are better than the Marlins. Therefore the chances the Nats gain a game in the standings is better than the chances they lose a game in the standings. It is more likely that we find the Nats 9.5 games up tomorrow than 7.5 games up (making a lot of assumptions - yes). Starting from this point it is more likely the Nats will be 12 up than 5 up after 5 games.  The streak analysis doesn't necessarily have to hold for this latter claim, teams can differ in how streaky they are, but it usually will, and in fact it does here. The Nats are pretty much exactly as likely to go 5-0 as the Marlins were to go 4-0, but the Marlins are way more likely to go 1-3 (or worse) than the Nats were to go 1-4. The end result is a 1.2% chance of the Nats expanding their lead to 12 games by this imperfect analysis.

By asking if the Nats are 5 games out a week or less from now would I have picked them at that point is essentially asking, "If you were to know an extremely rare and bad event would happen to the Nats in the next week, would you pick them" I might say no. But from this point I'll take the very very good odds that that extremely rare and bad event doesn't happen. Also note that when I made the call the Nats were in the beginning of a longish stretch against bad competition. The Marlins had just lost Stanton for the year. I didn't include that in any of the scenarios above but it is in there affecting all those odds in the Nats favor.

Donald also asks if maybe the Nats are not this good just on a hot streak now (and he doesn't ask but conversely let's go with the Marlins are better than this and are just cold) maybe they will streak to balance it out? He's right on the first points - though 120 games is a pretty good indicator, we don't really know the Nats true level. Part of that is just because that's a shifting target, with injuries, development, etc. But maybe they are a 92 win team instead of the 96 win team they are on pace for. (or the 102 win team RS/RA suggest or any other such projection) Who can say? And the Marlins could be better.

The problem is, for this worry, is that the "streak to even out" is not very likely. It doesn't work that way. Regression to the mean doesn't mean if you are a .500 team and you go 5-0 that you'll go 0-5 to get back to .500. It means that you'll play around .500 ball the more games you play and that winning percentage will drift back down to .500. 5-0 and a 1.000 winning percentage becomes 10-5 and 0.667, 20-15 and .571, 55-50 and .524.  If the Nats are worse and the Marlins are better, even if the Marlins are better than the Nats, it's hard for them to catch them because the most likely scenario over 20 games puts the Nats going say 11-9 and the Marlins 12-8, with the chances of different levels of success and failure rapidly diminishing from these central points. It's is very unlikely for a team, even if they are better to make up more than a few games over the course of a time frame even as long as a month.

Take the Nats - how have they done in a month? They've gained 1.5 games on the Marlins, 3.5 on the Mets, lost 2 games to the Phillies, and gained two on the Braves.  The Cubs have gone 20-8. That's huge! Back to their "maybe a top season of all-time pace" But even they didn't expand their lead by more than 4.5 games for three of the four teams behind them.

I called the Nats because it is simply very very unlikely for a Nats team that is better than the Marlins to find themselves losing all this ground in what remains of the season. Even if you don't think that the Nats are better, it's still very unlikely. Hell, even if you thought the Marlins were the best team in baseball and the Nats the worst, an 8.5 game lead over 45 games is pretty safe. It's over.


*This isn't the case, especially if you are looking at teams like the Marlins who are better than .500 but we're just spitballing here.