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The great thing about probabilities, the best thing, is that nobody short of God can ever prove you wrong. You say there's a 99 percent chance of rain? If it rains, you're right. If it doesn't rain, you're right. Just give a number greater than zero and less than one and you're good to go. 

On June 30, the Oakland A's were 13 games back of the Rangers, and our playoff odds measured their chances of winning the AL West at … zero. Zero percent. No chance. One of only two ways to have a provably wrong probability. Oh, but that's not all. June 30, the A's low-water mark in the standings, was one of 46 days that our playoff odds gave them a 0 percent chance in the West. The first was in April. The last was in July. 

Now, to be clear: Zero doesn't really mean zero. We round these things. Zero might mean zero, but it might just mean less than 1 in 2,001. (The Orioles chances of winning the division, if you're curious, have been at zero three days this year.) 

Those playoff odds may have been wrong, but they certainly weren't out of line with our words. For instance, there was episode 87 of Up and In, in which I suggested before the season that the A's wouldn't compete for perhaps five or six years, and oh wait it gets much much worse: 

Sam: A lot of A's fans, like one in 15 A's fans is actually going to be dead, gone from this Earth within five or six years. 
Kevin: That's bad math!
Sam: One in 15 adults is probably too much, yeah. But a large number of people are not going to be around to enjoy the next good A's team.
Kevin: "Many of their fans will be dead by the time they're good."
Sam: Many will be alive, though.

There was also this piece, written in June by resident A's expert Jason Wojciechowski, under the headline "Oakland Is Just Terrible (No Offense)." A's expert Jason Wojciechowski: 

Not that it really matters whether they hit better or not. Even if everyone regresses to what PECOTA thinks they're capable of, which involves a whole lot more regression upward than downward, their playoff odds as of this writing stand at 0.4 percent, up from 0.0 percent before Monday's game. The A's, 12-1 wins over the Rangers and near no-hitters thrown by phenom starters aside, are well and truly done.

In our preseason preview of that division, one of the questions about the A's was:

5) Can the A’s front-office survive a potential 100-loss season without losing its Hollywood shine? 

And one of the answers to that question was: 

The current version of the team isn't trying. If they lose 100 games, well, they traded all their good pitchers. Of course they lost 100 games! 

And it feels like a letdown at this point to remind everybody that our staff picks tabbed the A's as the likely last-place team in the AL West, with none of our 27 ballots picking them as a wild card team. I bring this up not to embarrass anybody; after all, I'm sure every one of us would have given the A's at least a one-in-a-million shot at the division, so every one of us totally called it. But in a few hours a pretty unexpectedly special game is going to happen, and so here's your unnecessary reminder of just how unexpectedly special it is. 

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Zero is a bad idea, short of mathematically eliminated.
If I believed in the possibility of such things, Sam, I'd say your "Many Of Their Fans Will Be Dead" bit is why the A's are in the playoffs, on the verge of a division crown.

That title is in the discussion for the 10 best "Up and In" episode titles of all time, BTW.

Team X "were 13 games back of the Rangers, and our playoff odds measured their chances of winning the AL West at ... zero." - doesn't this tell you that your formula needs some work?

And, how about some analysis on the Orioles and A's? Such as...

For the Orioles, tremendous fortune in 1-run games, and especially, extra-inning games.

For the A's, it's been very good pitching combined with unexpectedly great home run hitting. (The A's, post-All-Star, lead the majors in home runs. Did they even hit a home run in 2011?)
This is a key point and I think a huge flaw in many of the projections/odds for rest of season outcome, including some of mine.

The question you're answering is if we say the A's are a ~.500 team and the Rangers are a ~.580 team, what's the change the A's make up 13 games over the next 80 games. And the answer to that is indeed zero. That's the right question for making up 3 games over the last week -- because skill level is fixed and a .520 team is almost as likely to go 5-2 as a .570 team.

But over a long stretch, it's the wrong question. The question is what's the chance that the A's get significantly better (or are truly better) and/or the Rangers get significantly worse AND given that what's the chance of catching up. The A's weren't a .500 team that played crazy ball; the A's turned out to be a .600 team that played a bit better than expected. Not that anyone could have predicted that on June 30th.

In the past couple of years I've noticed a lot of teams playing large stretches where their "base skill level" is inconsistent with any analysis of their play so far. See the Pirates and White Sox late season collapses, or conversely the Padres run. These seem to happen WAY, WAY more often than simple probabilities would indicate.
I'm not sure if I've mentioned this here, but I'd love to see a study published on their own playoff odds. Just pick a day during the season, say June 10th, and see how accurate the odds are. A team with a 60% chance of making the playoffs that day should make it 60% of the time throughout the years they've tracked odds.

We seem to have so many historic comebacks/collapses recently - A's this year, Red Sox and Braves last year, the Mets not too long ago for two seasons in a row. Either these feats ARE incredible feats with little chance of happening, or the model could use a few adjustments.

I feel the model is sometimes a little too quick to bury a team or declare a team with a high probability of making the playoffs. Here are some examples from this season only: The White Sox had an 82% chance of making the playoffs in late May. That just seems too early, to me, for a team's playoff odds to be that high. The Dodgers were at 80% on May 22. The Angels were at 86% on July 2. Oakland was at 4% on July 13. Baltimore was at 8% on August 23.

That's 3 teams with an 80%+ chance to make the playoffs that didn't. I'm not saying the model is incorrect. Baseball is just an unpredictable game. But I'd just love to see some analysis on this, because there have been just enough problems and changes in the playoffs odds this season to make me a little wary in believing them.
I also seem to remember a BP article indicating that the acquisition of S Drew would make no difference in their playoff drive, when in fact he (and the platoon of Pennington and Rosales at 2B) seemed to strengthen their lineup quite a bit.
0.5 WARP in 38 games as an Athletic. not earth-shattering, but a pretty clear upgrade over the previous MI configuration.
Out of curiosity, what was the highest TEX's division odds ever reached?
"We round these things. Zero might mean zero, but it might just mean less than 1 in 2,001."

Since zero is absolute, until a team is mathematically eliminated their chances should never be rounded down.

I strongly suggest they always be rounded up to 1% in the future, or at least .0001%.