By Harrison Chase

Late yesterday, CBS Sports put out an article claiming that the Patriots were winning coin flips at “an impossible rate”. What’s more, they publicized it with the lead: “Have the Patriots found a way to get an edge on coin flips?” Clearly with Deflategate out of the way the media is looking for something to accuse the Patriots of.

Besides the fact that accusing anyone of cheating at coin flips is absurd, is the probability of this happening really that low? As said in the article, the Patriots have won 19 of their 25 past coin flips. The probability of winning it at least 19 out of 25 times is 0.0073, which is the number upon which they base their declaration of “impossible”.

But how impossible is it? Really, we are interested in not only the probability of getting 19 or more heads but also a result as extreme in the other direction – i.e. 6 or fewer. That probability is just 2*0.0073, or 0.0146.

That is still very low, however given that there 32 teams in the NFL, the probability of any one team doing this is much higher. To do an easy calculation we can assume that all tosses are independent, which isn’t entirely true as when one team wins the coin flip the other team loses. The proper way to do this would be via simulation, but assuming independence is much easier and should yield pretty similar results. The probability of any one team having a result that extreme, as shown before, is 0.0146. The probability of a team NOT having a result that extreme is 1-0.0146 = 0.9854. The probability that, with 32 teams, there is not one of them with a result this extreme is 0.9854^{32} = 0.6245998. Therefore, with 32 teams, we would expect at least one team to have a result as extreme as the Patriots have had over the past 25 games 1- 0.6245998 = 0.3754002, or 37.5% of the time. That is hardly significant. Even if you restricted it to not all results as extreme in either direction but just results of 19 or greater, **the probability of one or more teams achieving that is still nearly 20%.**

In addition the selection of looking at only the last 25 games is surely a selection made on purpose to make Belichick look bad. Why not look throughout his career? Did he suddenly discover a talent for predicting the future? Furthermore, given the length of Belichick’s career, we would almost expect him to go through a period where he wins 19 of 25 coin flips by random chance alone. We actually simulate this probability. Given that he has coached 247 games with the Patriots, we can randomly generate a string of zeroes and ones corresponding to lost and won con flips respectively. We can then check the string for a sequence of 25 games where there was 19 or more heads. I did this 10,000 times – **in 38.71% of these simulations there was at least one sequence with 19 or more heads out of 25.**

Anyways, both common sense and statistics will tell you that the Patriots have not been cheating by winning coin flips at an “impossible” rate. To be fair, the author of this article did not seem to insinuate that the Patriots were cheating, rather he was just remarking that it was a rare event (although, in reality, it shouldn’t be as unexpected as he makes it out to be). The fault seems to rather lie with who made the headline and pubbed it, although their job is probably just to get pageviews in which case I guess they succeeded.

Great post. I stumbled across it after doing most of the writing of my own. Hope the analysis pays appropriate homage: I played with a quick and dirty relaxation of the independence assumption.

http://www.mathofpolitics.com/2016/01/25/the-patriots-are-commonly-uncommon/