Arrests in the NFL versus the general population

I saw, on FB, this article on how a single NFL team has had no players arrested in the last 3 years, and since I'm a statistician, my first thought was, "how does that compare to the general population?"

Very back-of-the-envelope speaking, there were about 3.2 million arrests of males between the age of 21-34 (inclusive) in the U.S. in 2012**.  There were about 32 million males in that age range in the U.S..  So, very roughly speaking, about 0.1 arrests per year per male aged 21-34, which is roughly the demographic for NFL players.

Assuming that arrests occur like a Poisson process and using 0.3 arrests per 3-year interval as the parameter of a Poisson distribution, there is approximately a 74% probability of a randomly selected male aged 21-34 not being arrested in a three-year period.  The probability of 50*** randomly selected men all not being arrested is 0.74^50, or 0.00003%, so the probability that at least one person on that team was arrested is 99.99997%.

Thus, the very back-of-the-envelope estimate of the probability of all 32 NFL teams having at least one player arrested during a three-year period, if they were equivalent to members of the general public, is 0.9999997^32 = 99.999%.  So it seems to me that it's extremely unusual for *any* of the NFL teams to be arrest-free over a three year period.


** Click National Estimates, Annual Tables, 2012, Offense by Age for Males

*** 50 being approximately the size of an NFL roster