Preface: I'm probably wrong on the math below.
The second round of tests being PCRs and having a series of false positives is a lot more believable than there being > 1 false positive on the first round of lateral flow tests; we'll never get complete information, but I wanted to explore the likelihood of 2+ false positives from the same batch of lateral flow tests. A possible scenario:
- A set of players returns positive lateral flow tests (IIRC the initial reports said 3 players?) - 0.5% false positives among symptomatic subjects, 1.1% false positives among asymptomatic subjects according to
this resource. Assuming that none of the subjects that tested positive were symptomatic, and knowing TAA was actually infected, we're talking about two likely false positives among a test pool of ~30? Odds of 2 false positives in a row are ~0.0121%, 0.011*0.011/(1-0.011)^30 = 0.0168% chance of 2 false positives being incurred among any set of 30 tested players / staff using LFTs.
- A couple of assumptions - I imagine these tests are occurring daily across all 20 squads, and have been carried out for ~150 days (I know they changed testing standards recently and am lazy) - that means this 0.0168% chance has 3,000 independent opportunities of coming to fruition; unless I am completely misunderstanding probabilities (which, I cannot stress enough, is the likely outcome), it seems there's an over 50% chance of there being exactly 2 false positives on a set of 30 LFTs, if the same tests are executed 3,000 times. Obviously, there are an untold number of other variables that I'm not accounting for there, but when the testing volumes get to where they are these days, it's at least possible.
- The part that seems to introduce doubt about Liverpool's version of the story is arguably the more believable one, i.e. that the second round of tests had the same amount of (or more) false positives - from what I could find, albeit likely outdated (Jan '21 publish date),
this paper indicates that false positive rates in PCR testing can be as high as 5%. With ~30 tests, 0.05*0.05/(0.95)^30 yields a 1.16% chance of exactly two false positives - almost 100 times more likely than the LFTs. The combined probability of 2 false positives out of 30 in a series of LFTs, followed by 2 false positives out of 30 in a series of PCR tests, is about a 0.0000197% chance when each series of tests is executed once. Increase that by however many thousand instances of full teams being tested day-over-day, and you'll get a much larger number.
Is something up? Possibly - though I think we need more clear information on the volume of positive tests per testing cycle, time between tests, players who tested positive in each round (i.e. did the LFTs capture end-of-infection positives for one series of players, who then tested negative on the PCRs with other false positives cropping up?) etc.
I will say, I have no love for FSG - I didn't renew my club membership in light of the Super League bullshit, I think their model of finding the right talent to maximize their bang-for-buck is not suited to title chases against oil wealth, and that the club's recent success can be attributed to Klopp and Edwards, etc more so than any otherworldly ownership benevolence - but they are far greedier than they are stupid. I have a really hard time believing that an ownership group that's already taken a big black eye in the last 12 months would risk something so brazen and short-sighted - especially given this would not even get them out of any type of fixture congestion! Pushing the second leg back a week now just means the FA Cup tie against Cardiff is in limbo - but they haven't historically placed much emphasis on it anyway.
EDIT: Forgot to calculate out w factorials given finite number of tests per run. Oh well. Like 67 says, if it’s deliberate kick them out of the competition.