It Happens Once Every 250,000 Septembers

by J-Doug on Sep 29, 2011 5:15 PM EDT in Research

There's plenty of talk on the Twitterverse, in the blogosphere, and around other awkward neologisms about how improbable just last night was. As our own Satchel Price noted, last night included two Nolanesque comebacks-within-a-comeback. It's the unlikeliness of the larger comeback (and collapse) that I'd like to share with you here.

It's one thing to discuss this past September in the context of the playoff chances of individual teams. But for both St. Louis and Tampa Bay to make it? Well, the numbers seemed nigh on impossible. Posted below is a log-scale chart plotting the likelihood of both underdogs reaching the playoffs, as well as the chances of both favorites missing October.

Click to enlarge. Data available after the jump.

This chart has to be in log-10 scale. There is no other way to present the improbability of a Sox-Braves joint collapse or Rays-Cards joint comeback. Less than a week ago, the odds of either outcome occurring were closer to 1 in 1000 than 1 in 100. On September 4th, the odds for either joint-outcome occurring were a dismal 1 in 100,000. Two days later, the odds of both Tampa Bay and St. Louis playing October were a mind-boggling 1 in 250,000.*

Eh, what the heck. Here's the same chart on a typical base-10 scale:

I suppose that better portrays the drama of this past September, if not the finer details of what we all witnessed.

Of course, the real odds on 9/6 of the Tampa Bay Rays and St. Louis Cardinals reaching the playoffs were probably better than 0.0004%. A proper Bayesian would update what he knew before with what he knows now. But even if we appropriately regress this figure toward the historical commonality of this race occurring (just under 1%) the "true" improbability would still be rather staggering.

As those in the mainstream media recounted last night's events--and the context that framed them--they used phrases such as "once-in-a-lifetime" or "won't-ever-happen-again." Well, the numbers agree. This certainly hasn't happened before, and it probably won't happen in our lifetime, or in many, many lifetimes following ours.

Apologies to Sox and Braves fans, but this is one for the ages.

Date	BOS Reaches	TBR Reaches	ATL Reaches	STL Reaches	BOS+ATL Miss	TBR+STL Reach	Notes
9/1	99.9%	0.2%	98.6%	3.3%	0.0014%	0.0066%
9/2	99.9%	0.1%	98.6%	1.5%	0.0014%	0.0015%
9/3	99.9%	0.1%	97.8%	1.7%	0.0022%	0.0017%
9/4	99.9%	0.1%	99.0%	0.9%	0.0010%	0.0009%
9/5	99.7%	0.2%	98.8%	0.7%	0.0036%	0.0014%
9/6	99.9%	0.0%	98.1%	0.9%	0.0019%	0.0004%	Coded TB as 0.04%*
9/7	99.9%	0.1%	97.2%	2.1%	0.0028%	0.0021%
9/8	99.7%	0.2%	99.0%	1.0%	0.0030%	0.0020%
9/9	99.3%	0.7%	98.3%	2.2%	0.0119%	0.0154%
9/10	98.7%	1.2%	96.9%	4.0%	0.0403%	0.0480%
9/11	97.1%	2.8%	94.4%	6.2%	0.1624%	0.1736%
9/12	96.0%	4.0%	95.2%	5.1%	0.1920%	0.2040%
9/13	98.1%	1.8%	95.5%	4.5%	0.0855%	0.0810%
9/14	97.6%	1.7%	96.1%	4.2%	0.0936%	0.0714%
9/15	95.1%	3.8%	96.0%	4.2%	0.1960%	0.1596%
9/16	98.1%	1.6%	90.4%	8.8%	0.1824%	0.1408%
9/17	95.3%	4.5%	95.5%	3.4%	0.2115%	0.1530%
9/18	88.1%	10.9%	90.8%	6.5%	1.0948%	0.7085%
9/19	85.8%	13.7%	79.9%	16.9%	2.8542%	2.3153%
9/20	83.0%	15.2%	81.0%	18.3%	3.2300%	2.7816%
9/21	82.8%	11.0%	72.7%	26.0%	4.6956%	2.8600%
9/22	85.5%	12.5%	84.5%	15.0%	2.2475%	1.8750%
9/23	94.6%	4.1%	95.4%	4.6%	0.2484%	0.1886%
9/24	87.5%	9.0%	88.8%	11.2%	1.4000%	1.0080%
9/25	89.8%	9.8%	71.7%	28.3%	2.8866%	2.7734%
9/26	54.9%	45.1%	78.7%	21.3%	9.6063%	9.6063%
9/27	48.8%	51.2%	45.6%	54.4%	27.853%	27.853%	Based on single game sims.
9/28	0.0%	100.0%	0.0%	100.0%	100.00%	100.00%

Data courtesy of Baseball Prospectus: http://www.baseballprospectus.com/odds/

*Baseball Prospectus lists Tampa Bay's chances of reaching the playoffs following the games on 9/6 as 0.0%. This is a rounding issue. Based on the wild card chances of the other AL teams, Tampa Bay's simulated playoff chances were at best 0.04%, but perhaps less. In other words, 1 in 250,000 might have been a tad optimistic.

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I like how the infinitesimally small odds were clearly displayed. I still can’t believe what happened. That being said, I still wish they had tied (all four teams) and had to play games 163

by bloppy_ploppy on Sep 29, 2011 9:56 PM EDT reply actions

Thanks!

And I agree. I was among the Yankees fans rooting for a Rays victory, but once the Orioles walked off I suddenly felt conflicted. For about three minutes. Then Evan Longoria solved my problem.

Blogger and Editor, Rational Pastime Blog. Twitter: @RationalPastime.

by J-Doug on Sep 29, 2011 11:14 PM EDT up reply actions

Unreal.

I get to tell my children about this someday.

Wear your own fur.

by Marc Fournier on Sep 29, 2011 10:11 PM EDT reply actions

+1

Blogger and Editor, Rational Pastime Blog. Twitter: @RationalPastime.

by J-Doug on Sep 29, 2011 11:14 PM EDT up reply actions

So either we just witnessed an event that had a 1 in 250,000 chance of happening or

the method of computing the probability of such a thing is flawed. Thinking like a Bayesian I have to say I favor the latter explanation.

Let me ask you this. When you look back at all of the biggest ‘collapses’ in major league history and you look at the probability of each one (computed based on the same methodology used above), do we see the number of collapses that this method of probability predicts we will or do we see far more? I think we see far more.

Just like Nassim Taleb stated, we see far more financial collapses than most models predict that we will. I believe that the reason is simple. These probability models treat each game as a completely independent event. They are not. One non-psychological factor is bullpen usage. Losing games generally leads to more bullpen usage which leads to bullpen fatigue which leads to worse bullpen performance which leads to more bullpen usage etc…. Then there is the pressure that comes with trying to protect a shrinking lead in the standings. This is easy to understand intuitively. Continued losing leads to more pressure which leads to impaired performance and more losing etc..

To say it another way we did NOT just witness a 1 in 250,000 sequence of events. But feel free to bet on those odds next time a time is up 9 games with 30 to play.

by TMS71 on Oct 5, 2011 3:10 PM EDT reply actions

Did you bother to read this paragraph before you wrote that dissertation?

Of course, the real odds on 9/6 of the Tampa Bay Rays and St. Louis Cardinals reaching the playoffs were probably better than 0.0004%. A proper Bayesian would update what he knew before with what he knows now. But even if we appropriately regress this figure toward the historical commonality of this race occurring (just under 1%) the “true” improbability would still be rather staggering.

Blogger and Editor, Rational Pastime Blog. Twitter: @RationalPastime.

by J-Doug on Oct 6, 2011 1:11 AM EDT up reply actions