When the Panthers’ safety spoke out about how often he had been drug tested since signing with Carolina, we enlisted the help of a Fairfield professor to help us calculate the percent chance of being drug tested six times in 11 weeks.

CHARLOTTE — Eric Reid is posing for a photo in the locker room about a half-hour after the Panthers’ 12–9 loss to the Saints on Monday Night Football.

He’s shirtless, wearing team-issued gray shorts, standing in front of a blank wall in the threshold between the shower area and locker room. He holds his cell phone in his hand as his Twitter mentions blow up. The man holding his own phone horizontally and snapping a headshot of Reid will soon collect Reid’s urine specimen for testing.

Again.

According to Reid, this is the seventh time he’s been drug tested since signing with the Panthers in late September, during Carolina’s bye week. Allowing for one mandatory drug test (which Reid might have taken in the offseason if he had been on an NFL roster; he’s also never tested positive, so he’s not subject to “probable cause” testing) the Panthers safety claims this is the sixth random test he’s been required to take in 11 game weeks.

By noting the alleged frequency of the tests, Reid is clearly calling into question the “randomness” of the league’s drug-testing system, at least as it relates to him. He and many NFL observers are dubious, and some believe more nefarious factors are afoot as they relate to the league and one of its most outspoken players—one who happens to be suing the league.

After providing a speciman on Monday night, Reid pulled on a black t-shirt reading BRING BACK KAP. Reid is Colin Kaepernick’s closest friend in the NFL. He was the first player to join Kaepernick in kneeling during the national anthem to protest social injustice and racial inequality in America. In the spring he filed a grievance with the league, via the NFLPA, alleging that NFL owners and the league colluded to prevent his signing, due to his protests and to pressure from the influence of President Donald Trump.

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Is it statistically impossible that Reid has been randomly drug tested six times in 11 weeks? Certainly not. Is it statistically improbable? Absolutely, yes.

“There’s a .195% chance of flipping heads nine times in a row,” says Nick Kapoor, an adjunct professor of mathematics at Fairfield University. “You have [about] the same probability of flipping a heads nine times in a row as he is being tested six times in 11 weeks.”

I reached out to Kapoor on Tuesday afternoon because of his background and because I’m terrible at math. Kapoor, a vice president of a software company, teaches statistics and calculus at Fairfield. He holds a master’s degree in mathematics from the school and an MBA from Sacred Heart University. In 2016 he won $100,000 playing Powerball when he matched numbers on four white balls and the bonus ball. (Despite his background in statistics, he did not “game” the lottery. Winning Powerball is known in statistics as an independent event, and his 1-in-913,000 odds were the same as yours that day.)

“I’ve always played the lottery. It’s so funny because I teach my students about probabilities and what it means to win, and the academic side of me says, ‘Don’t play the lottery because you see the numbers,’” Kapoor says. “But then the small, optimistic part of you says someone’s got to win eventually.”

Kapoor didn’t know who Reid was before we spoke. I explained the situation and the league’s drug-testing policy, and he calculated that there’s a 0.169% chance that a player would be randomly selected for a drug test six out of 11 weeks.

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The league’s process for testing for prohibited substances is laid out in the 2011 collective bargaining agreement. It’s a joint policy between the NFL and the NFL Players Association, and the testing is overseen by an independent administrator—for the past quarter-century that’s been Dr. John Lombardo. He’s paid by both the league and the union, and the two entities have the power to fire him as well.

Players are selected randomly by a computer program each week. Names are not used in this system. Rather, every NFL player has his own unique player ID, kind of like an employee number like many of us have. Ten players from every team—drawn from the active roster, practice squad and reserve list—are selected each week. The tests are performed throughout the week, some immediately after a game and others in the middle of the practice week.

“Players will be required to provide a specimen whenever they are selected, without regard to the number of times they have previously been tested,” the policy states.

This practice may seem annoying or like overkill, but there’s a practical reason for it. Players must always go back into the pool of testing candidates in order for the process to be effective. If the league says it will randomly test players until they hit, say, four tests, theoretically once a player hits that number he could pop steroids like a 1980s pro wrestler for the rest of the season, knowing he’s been taking out of the pool for random testing. A league source told me they have seen players go all year without a test, and players tested double-digit times in one season.

Finding the chances of being randomly selected six times in 11 weeks required Kapoor to follow a binomial distribution. To adhere to that, each week must be independent, there must be only two possible outcomes (either the player is tested or he is not) and the probability of being chosen in any given week is constant. In order to keep the math relatively simple, we set our sample roster at 71 players. That’s 53 active players, 10 practice squad players and eight players on reserve. (The Panthers had 76 players on Monday night, but they would have had fewer players on reserve in Week 5, the first game after Reid signed, so we sought a reasonable average to find a constant roster number.)

The mean of this binomial distribution is 1.549. That means a particular player should be expected to be randomly chosen 1.549 times out of 11 weeks—between once and twice. Six times in that span is a clear statistical outlier.

There are really only three possible explanations or outcomes for this situation:

1. Reid is mistaken or, worse, lying about how many times he’s been tested.

2. The league has rigged the system, presumably without the knowledge of the players’ union, to make Reid’s tests not random but deliberate.

3. Reid is defying the odds and is in that 0.169% of people who are chosen six out of 11 weeks.

Regarding scenario 1, Reid could settle the debate by publicly furnishing proof of his claim, if he has retained it. To do so would risk showing his hand while he has an ongoing lawsuit against the league, and he’s been very careful in the past three months not to reveal too many details related to his grievance. And while the league and the union have access to his testing information, neither can release an individual’s medical records without his consent. Even if the union could disclose the information, confirming Reid’s account might harm his grievance, and discrediting him would make the union look like traitors in the eyes of the players.

But let’s say he was mistaken about one of the tests. Not that of the seven, one was mandatory, but two were—one for recreational substances, and a second for performance enhancing drugs. According to the math, there’s a 1.03% chance of being randomly tested five times in 11 weeks. Don’t want to give him the benefit of the doubt? Think that he is flat-out lying about the number of tests? OK. There’s a 4.5% chance of his being randomly tested four times in 11 weeks, and a 13.7% chance of being tested three times in that span. Believe what you will, but for the sake of consistency, I will stick to his claim of six random drug tests in 11 weeks for the remainder of this story.

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Scenario 2 posits that this random testing isn’t so random, after all. The probabilities Kapoor spun out don’t mean much if things are being tampered with.

In this picture, the league is working in concert with a third party to test Reid more regularly than his peers. The reasoning here would be either to annoy and harass Reid enough to drop his suit against the league or to increase the chances of catching him using a prohibited substance, thereby discrediting him in his suit and harming his reputation in the public sphere.

The league tampering with the negotiated process would certainly lead to litigation. How would the NFL tamper, exactly? It’s unclear. Someone within the league with great influence might be leaning on a third party. To be clear, there is no evidence of tampering and I am not accusing anyone of doing such. But Reid indicated to reporters last month, shortly after his claimed fifth random test following the Week 12 loss to Seattle, that he wouldn’t put tampering past the NFL.

“I don’t have anything to hide [by being tested repeatedly],” Reid said, according to the Charlotte Observer. “Secondly, this is supposed to be a random system. It doesn’t feel very random. Plus, I’m privy to information that’s in my lawsuit that’s not free to the public. So I know who I’m going against and it’s not surprising in the least.”

Now let’s look at scenario 3. If you sit there long enough and keep flipping that coin, eventually, according to statistics, you’ll get nine heads in a row. Maybe that’s what happened here.

As much of an outlier as it would be for any particular player, it’s statistically possible—likely, even—that at least one player in the NFL this season would be chosen randomly six times in 11 weeks. Assuming the league is testing roughly 2,000 players, you’d expect between three and four players (.169% of 2,000 = 3.4) to be tested six times in 11 weeks.

That it happens to a particular player who also happens to be suing the NFL?

“Unusual? Yes,” Kapoor says. “Quite unusual? Yes. Improbable even? Yes. Impossible? No.”

In the past few weeks, Reid has noted the various low probabilities people have placed on this occurrence—that is, on the fact that someone tested so frequently would also happen to be engaged in an acrimonious grievance with the league. On Monday night, asked if this statistical improbability would be used in his suit against the NFL, Reid smirked, looked to his right and nodded slightly.

“Duly noted,” he said.

Question or comment? Email us at talkback@themmqb.com.