Probability and belief

A few days ago, in Probability and the Problem of Life, I pointed out that there is no need to precisely calculate probabilities that we cannot possibly know in order to reach logical conclusions about them. Contra the opinions of the misguided math fetishists, logic is the foundation of math, not the other way around, and we can reach perfectly sound logical conclusions even if we are not able to make precise mathematical determinations or quantifiy all of the various factors involved.

Throughout the course of the discussion, it soon became abundantly clear that those who defend the theory of TENS on probability grounds do not actually believe their own position. Furthermore, it is relatively easy to demonstrate that although the very low probability events to which they appeal are mathematically possible, they are so highly improbable that no sane human being can credibly feign to take seriously, as evidenced by their own daily behavior with regards to other, much more likely events.

WRF3 asked me to identify the precise point at which mathematical possibility and belief part company; I said that for me it was somewhere between 1 in 4,165 and 1 in 17,347,225. The latter are the odds of being dealt four aces twice in succession from two properly shuffled card decks; I would not view that as credibly possible and continue to play poker with a machine that dealt out such hands. The absolute outer limit for even the most credible individual is probably 1 in 72,251,192,125, which would be three such unlikely hands.

But the reality is that for the average individual, the credibility ratio is much lower. Consider the recent statistical evidence of the New England Patriots having systematically cheated by deflating the football since the 2007 season:

While speculation exists that “Deflate Gate” was a one time occurrence, data I introduced last week indicated that the phenomena MAY have been an ongoing, long standing issue for the New England Patriots. Today, that possibility looks as clear as day.

Initially, looking at weather data, I noticed the Patriots performed extremely well in the rain, much more so than they were projected.  I followed that up by looking at the fumble data, which showed regardless of weather or site, the Patriots prevention of fumbles was nearly impossible.  Ironically, both studies saw the same exact starting point:  2007 was the first season where things really changed for the Patriots.  Something started in 2007 which is still on-going today.

I wanted to compare the New England Patriots fumble rate from 2000, when Bill Belichick first arrived in New England, to the rest of the NFL.  Clearly, one thing I found in my prior research was that dome teams fumble substantially less frequently, given they play at least 8+ games out of the elements each year.  To keep every team on a more level playing field, I eliminated dome teams from the analysis, grabbed only regular season games, and defined plays as pass attempts+rushes+times sacked.  The below results also look only at total fumbles, not just fumbles which are lost.  This brought us to the ability to capture touches per fumble.

To really confirm something was dramatically different in New England, starting in 2007 thru present, I compared the 2000-06 time period (when Bill Belichick was their head coach and they won all of their Super Bowls) to the 2007-2014 time period.  The beauty of data is the results speak for themselves:

The data is jaw dropping, and this visual perfectly depicts what happened.  From a more technical perspective, John Candido, a Data Scientist at ZestFinance who is a colleague of mine over at the website and was also involved in the development of this research, comments:

Based on the assumption that plays per fumble follow a normal distribution, you’d expect to see, according to random fluctuation, the results that the Patriots have gotten since 2007 once in 5842 instances.

Which in layman’s terms means that this result only being a coincidence, is like winning a raffle where you have a 0.0001711874 probability to win. In other words, it’s very unlikely that results this abnormal are only due to the endogenous nature of the game.

Many of the arguments giving the Patriots the benefit of the doubt are evaporating.  While this data does not prove they deflated footballs starting in 2007, we know they were interested in obtaining that ability in 2006. (This is something I found out AFTER I performed the first two analyses, both of which independently found that something changed starting in 2007.)

I was skeptical when I first read the analyst’s theory, because he initially used fumbles lost rather than all fumbles; it is generally believed by football statisticians who have considered the question that fumble recoveries are random. And when fumbles rather than fumbles lost are utilized, the Patriots are considerably less of a radical outlier, although they are the only team that plays outdoors that fumbles as little as a dome team.

My first thought was that the anomaly was more a result of New England’s pass-happy offense than statistical evidence of ball deflation. However, a look at the passing statistics showed that New England was pass-happy as early as 2002, when they threw 601 passes, compared to 582 in 2014, and the fact that their plays per fumble from 07-14 increased so dramatically from 00-06 after the rule change that they requested does tend to confirm the analyst’s original suspicions.

 But my point is not to take a side in the latest New England scandal, only to observe that for the professional statistician, observation of a successful event against 1 in 5,842 odds is sufficient to indicate the results observed are probably not obtained naturally. And while this statistical evidence is not absolute proof (although it is interesting to see that the statistician’s odds are in the range I suggested should preclude belief), it is enough to indicate that the greater part of one’s efforts should be directed at discovering the precise nature and mechanism of the unnatural tampering indicated rather than on the unlikely natural explanation.

“The bottom line is, something happened in New England.  It happened just
before the 2007 season, and it completely changed this team.”

Which brings us back rather to my long-held position contra Mr. Sherlock Holmes: Once you have calculated the sufficiently improbable, you must reconsider your assumptions of the impossible.