So the Tigers lost to the Rays earlier today. I'm not a big fan of the "games behind" metric these days for a couple of reasons.
1) Being behind 65-65 to 69-61 is different than being behind 63-63 to 69-61, but by "games behind" it shows up as the same 4.0 games behind.
2) Because the schedule is a lot more unbalanced than it has been historically, this leads to a lot of naive statements like "only (small percentage) of teams who were behind by N games with D days left in the season made it to the playoffs," which are well-intentioned (a lot of people will overestimate their chances for making a comeback), but annoy me because the unbalanced schedule changes things.
Of course, just because I find something annoying doesn't mean that it is giving us a huge error in the calculation. How big of an effect does the unbalanced schedule have on this playoff race?
Right now, the Tigers stand at 8 games above .500, the Twins exactly at .500, and the White Sox at 3 games below .500. That makes it seem like there's a pretty big difference in talent between the teams, but they are all within 15 runs of an even run differential. It's completely within the realm of probability that the three teams could reverse their order in the run differential standings by the end of the season.
So I think that going forward, it's pretty fair to consider the teams roughly equivalent, and that the odds for any team to win those games is roughly 50%. Yeah, the Tigers are going to be slight favorites overall, since they have performed better, but they just haven't performed all that much better for me to think the difference is worth worrying about.
One naive way of simulating the rest of the season would be to say that each team has a 50% chance of winning each game the rest of the way out, independent of the other teams' games. (This is the underlying assumption made in statements like "if the Tigers go .500 the rest of the way out, the Twins would need to play 20-12 in September to make the playoffs.") If you do that, you get roughly the following odds of victory in the division:
79.5% -- Detroit
14.5% -- Minnesota
6.0% -- Chicago
By that way of figuring, the Tigers seem to have things pretty well wrapped up.
However, we've made a (rather technical) mistake here in assuming that all game results are independent. We have three groups of games where the results are 100% correlated:
7 games -- Tigers vs. Twins
6 games -- Tigers vs. White Sox
6 games -- Twins vs. White Sox
Instead of flipping 95 coins to play out the rest of the season, we should be flipping just 76 coins to play out the rest of the season. If we do that, then we get the following odds:
76.4% -- Detroit
16.4% -- Minnesota
7.2% -- Chicago
And we see that I'm probably more annoyed than I ought to be. The unbalanced schedule makes a difference, but not really a huge difference.
Anyway, the Twins are definitely in the race. A 16% chance is pretty small, but roughly corresponds to the frequency with which Matt Tolbert was hitting this year while he was up with the Twins. So these sorts of things happen, but you don't want to bet the farm on them.
P.S. It looks like coolstandings.com has us at around 20%, so we probably have some slight strength-of-schedule advantage in the non-contender games.
P.P.S. I calculated the odds using a Mathematica notebook, which I'm happy to share if there is any demand.
The odds went up after last night.
28.2% sounds better, but don't count Cleveland out -- coolstandings hasn't
The Clevelands have the best chance of any 4th-place team to make the playoffs!
I think that by giving Cleveland 1-in-500 odds, Coolstandings is in fact counting Cleveland out.
ubes = ice water
after two more games against the White Flags, I might start to get excited.
the season ends with 3 vs. Det, 3 at the White Flags, 3 at KC, and 3 at home vs. Det. I see all kinds of possibilities for excitement and heartbreak.
whoops. there's October baseball two (err, too). Make that FOUR at home vs Det, then a final 3 vs. the Gentlemen.
ZOMG! Imagine going into that final 3-game set within a game of Detroilet!!!!111one11!!!!
Mathematica? gross!!! I f'n hate that program with a fiery burning passion (too many "you have a syntax error somewhere in the triple integral, but I'm not going to tell you where, Ha Ha!" errors for me). I swing the matlab (or in a pinch, the TI-89) way.
Mathematica is gross, but I know it and it gets the job done. Also, our department has access to it and not Matlab, and I don't feel like breaking the law or the bank for the privilege of learning some new syntax that basically does what I can already do with Mathematica.
That's fair enough. I just have awful memories of it from my early undergrad. days and feel obligued to bash it whenever possible. I know it's an amazingly powerful tool, but the syntax errors are just so vexing. I would invariably put a curly bracket instead of a square and end up spending a half hour poring over every character.
It's not really just the syntax errors--the syntax is awful. Well, it's awful if you want to try to do something that accesses particular elements of a vector or matrix, especially since they used to not have any syntax highlighting at all. Stuff like a[[b,c]] quickly turns into a[[ b[[i]],c[[j]] ]] and even with some reasonable whitespace gets to be unreadable in a hurry.
I think that one reason the syntax starts to get so awful is that Stephen Wolfram wants you to use functional programming rather than imperative programming, and most of us are more used to the latter, I think.
That is certainly true. My biggest beef is that the program could be set up so that the offending bracket could just be highlighted with a responible error message.
Matlab really is a lot easier to use because it shamelessly borrows from any and all programming languages (C, C++, BASIC, FORTRAN, etc) so if you know how to program, you can get it to do exactly what you want. It can't do the same level of heavy lifting as Mathematica, but it is far more versatile.