Ok, I thought this was interesting and wanted to see if anyone could figure it out cause the media can't.
Here is the situation, Eric Reed is an NFL player, He has been in the NFL for 11 weeks making him subject to 11 rounds of "Random" drug test.
-He has been selected for 6 of the 11 possible test. -There are approximately 72 testable players on a team and each week 10 are selected at random. -There are 32 teams in the league.
So how would you go about finding the probability that this player was not chosen randomly but was targeted.
Also please feel free to expand on this question in anyway you feel is appropriate as this is not my wheelhouse and I'm sure you could change the question to be more meaningful and statistically significant if you wanted to and that's what I'm really after... this will be linked to in a few forums for people who don't necessarily use math lingo so please feel free to draw conclusions or give simplified perspectives.
Thank You!!
If everything is uniformly random (which need not be the case), then there are $32\cdot 72 = 2304$ total players in the pool for random testing; since $10$ are chosen each week, the probability that a player is not chosen to be tested is $$ \frac{2303}{2304}\cdot\frac{2302}{2303}\cdots\frac{2294}{2295} = \frac{2294}{2304} $$ so the probability that the are chosen is $1-\frac{2294}{2304}=\frac{5}{1152}$. Call this value $p$.
Now we are in a classic Bernoulli trial setup; we have a probability of success (being called for a drug test) and a number of trials (each of the 11 weeks that have passed), and we wish to know the probability that we succeed a certain number of times. Plugging everything in we find that the probability of being called at least $6$ times would be $$ \sum_{k=6}^{11}{11\choose k}p^k(1-p)^{11-k} \approx 3\cdot 10^{-12} $$ which is a touch on the low side.
What does this mean? Either nothing, or not that much, or it is the scandal of the century. First, everything that is possible is possible (deep wisdom I know). It could be that this player was (un)lucky enough to win this lottery. Second, the procedure may not be uniformly random. While it might strike you as "unfair" to weight some players as riskier than others, actuaries do this all the time with things like drug use, recidivism rates, life expectency, etc. It might not be a stretch to think that the NFL also has a different distribution than uniform randomness for these drug trials. You can ask why they should deviate from uniform randomness, but that would at least help explain why such a "rare" event has happened. And third, it could be that it isn't really random, or someone messed with the system to circumvent randomness, and we have the next scandal on our hands that we can fret over for 24-72 hours.
EDIT: Based on a comment on this answer, I did some quick research and discovered that the testing is not NFL wide as I have assumed, but that each team tests $10$ of their players each week. This gives you a different $p$ value of $5/36$, and the rest of the numbers change accordingly.