Last November, CBS Sports caused a tempest in a teapot with an article with the sensational headline “Patriots have no need for probability, win coin flip at impossible rate.” From the opening paragraphs:

Bill Belichick is never unprepared. Or at least that’s the perception. When other coaches struggle with when to use timeouts or how to manage the clock, the Patriots coach, almost effortlessly, always seems to make the right decision.

Belichick has also been extremely lucky. The Pats have won the coin toss 19 of the last 25 times, according to the

Boston Globe‘s Jim McBride.For some perspective: Assuming the coin toss is a 50/50 proposition, the probability of winning it at least 19 times in 25 tries is 0.0073. That’s less than

three-quarters of one percent.

As far as the math goes, the calculation is correct. Using the binomial distribution,

.

Unfortunately, this is far too simplistic an analysis to accuse someone of “winning the coin flip at an impossible rate.” Rather than re-do the calculations myself, I’ll just quote from the following article from the Harvard Sports Analysis Collective. The article begins by noting that while the Patriots may have been lucky the last 25 games, it’s not surprising that some team in the NFL was lucky (and the lucky team just happened to be the Patriots).

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%.

The article goes on to note the obvious cherry-picking used in selecting the data… in other words, picking the 25 consecutive games that would make the Patriots look like they were somehow cheating on the coin flip.

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.

The author makes the following pithy conclusion:

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.

At any rate, the Patriots lost the coin flip in the 26th game.