That calculation is pretty straightforward using the binomial theorem, except ... the binomial theorem requires all events to be independent of each other, i.e. the chances of one person getting a false positive is not affected by the chances of another person getting a false positive. But that is almost certainly not the case here - it's much much much more likely that there was some kind of cross contamination.I know I made some rough estimated predications yesterday and earlier today about how unlikely multiple false positives on a Lateral Flow Test might be. Now I have finally/sadly crunched the numbers for real, it seems I actually over-estimated the likelihood of multiple falsely positive Lateral Flow Tests occurring at Liverpool recently.
I appreciate I've been banging on about this, but just so everyone is clear (those with lives look away now) here is my working:
The United Kingdom Health Security Agency (UKHSA) has made it clear that a positive indication on a lateral flow test is correct 9,997 times out of 10,000 (that is, subsequent PCR tests have proved them to be correct). In fact, they are so reliable (from a positive test point of view at least) that producing one has removed the need to take any further confirmatory test (e.g. a PCR) as was previously government policy.
The probability of suffering a false positive on a LFT is 3 in 10,000. I read today that the UKHSA are saying that, so far as the new Omicron version goes, getting a false positive for that strain is ten times even less likely than it was for Delta i.e. the probability, i.e. occurrence, of false positives on LFT's for Omicron is a vanishingly small 3 in 100,000.
For those that care, calculate the probability of Liverpool players encountering just three false positives in a row when taking an LFT from those in circulation in this country.
That is (3/10000)*(3/10000)*(3*10000)
i.e. (0.0003) x (0.0003) x (0.0003)
That comes to 2.7 times in 100 billion i.e. 2.7 times such a scenario would arise in 100,000,000,000 times of trying it.
To get 15 false positive lateral flow tests on the bounce you would multiple the chance of it occurring just the once i.e. 0.0003 and that happening on 14 subsequent occasions. You would multiple 0.0003 by the probability of producing another false positive LFT test which remains the same (i.e. 0.0003) and do so with the results of that sum a further 14 times. That is 0.0003 to the power of 15.
It produces such a small chance of probability (it has 54 0's after the decimal point before you get 1435) that it is the equivalent of every star in the universe (higher estimate to be 10^24 which is 1 with 24 zero's after it) being multiplied by the same number, and then the produce of that little lot multiplied a mere 10,000 further times for further good measure.
There is a 1.435 chance in 10 to the power 54 chance (a 1 with 54 zeros after it) of drawing out 15 false positive randomly selected Lateral Flow Tests on the bounce.
I appreciate that not all 15 had to be on the trot and that they could play 9 jokers (no false positive tests) in amongst those to make up the squad of 24. I will let a better man than me work out this precisely chances of encountering 15 false positive LFTs out of a squad of 24, it's clearly slightly less than I have indicated with the above illustration. It's a long time know since I did O' level maths. You did your best Mr Meakin!
So, if the batches of lateral flow tests did not prove to be faulty, Liverpool might simply have just been a little unlucky in this incidence. If it's often the case with them I hear.
That calculation is pretty straightforward using the binomial theorem, except ... the binomial theorem requires all events to be independent of each other, i.e. the chances of one person getting a false positive is not affected by the chances of another person getting a false positive. But that is almost certainly not the case here - it's much much much more likely that there was some kind of cross contamination.
So, put simply,I know I made some rough estimated predications yesterday and earlier today about how unlikely multiple false positives on a Lateral Flow Test might be. Now I have finally/sadly crunched the numbers for real, it seems I actually over-estimated the likelihood of multiple falsely positive Lateral Flow Tests occurring at Liverpool recently.
I appreciate I've been banging on about this, but just so everyone is clear (those with lives look away now) here is my working:
The United Kingdom Health Security Agency (UKHSA) has made it clear that a positive indication on a lateral flow test is correct 9,997 times out of 10,000 (that is, subsequent PCR tests have proved them to be correct). In fact, they are so reliable (from a positive test point of view at least) that producing one has removed the need to take any further confirmatory test (e.g. a PCR) as was previously government policy.
The probability of suffering a false positive on a LFT is 3 in 10,000. I read today that the UKHSA are saying that, so far as the new Omicron version goes, getting a false positive for that strain is ten times even less likely than it was for Delta i.e. the probability, i.e. occurrence, of false positives on LFT's for Omicron is a vanishingly small 3 in 100,000.
For those that care, calculate the probability of Liverpool players encountering just three false positives in a row when taking an LFT from those in circulation in this country.
That is (3/10000)*(3/10000)*(3*10000)
i.e. (0.0003) x (0.0003) x (0.0003)
That comes to 2.7 times in 100 billion i.e. 2.7 times such a scenario would arise in 100,000,000,000 times of trying it.
To get 15 false positive lateral flow tests on the bounce you would multiple the chance of it occurring just the once i.e. 0.0003 and that happening on 14 subsequent occasions. You would multiple 0.0003 by the probability of producing another false positive LFT test which remains the same (i.e. 0.0003) and do so with the results of that sum a further 14 times. That is 0.0003 to the power of 15.
It produces such a small chance of probability (it has 54 0's after the decimal point before you get 1435) that it is the equivalent of every star in the universe (higher estimate to be 10^24 which is 1 with 24 zero's after it) being multiplied by the same number, and then the produce of that little lot multiplied a mere 10,000 further times for further good measure.
There is a 1.435 chance in 10 to the power 54 chance (a 1 with 54 zeros after it) of drawing out 15 false positive randomly selected Lateral Flow Tests on the bounce.
I appreciate that not all 15 had to be on the trot and that they could play 9 jokers (no false positive tests) in amongst those to make up the squad of 24. I will let a better man than me work out this precisely chances of encountering 15 false positive LFTs out of a squad of 24, it's clearly slightly less than I have indicated with the above illustration. It's a long time know since I did O' level maths. You did your best Mr Meakin!
So, if the batches of lateral flow tests did not prove to be faulty, Liverpool might simply have just been a little unlucky in this incidence. If it's often the case with them I hear.
They never get investigated for anything. Take the bus incident, nobody arrested. The spitting incident earlier this season, they looked into it , nothing. And don’t get me started with the hacking of our computer systems. Teflon scousers.So, put simply,
Chance of Liverpool being dirty filthy cheats =100%
Chance of Liverpool being found guilty of anything = 0%
Pretty sure my maths is correct.
I recognise your reasoning - at least in view of the assumptions I made in my earlier number crunching - where I tried to illustrate the unaffected chances of one person randomly receiving a false positive, which was turn followed by another etc, etc.That calculation is pretty straightforward using the binomial theorem, except ... the binomial theorem requires all events to be independent of each other, i.e. the chances of one person getting a false positive is not affected by the chances of another person getting a false positive. But that is almost certainly not the case here - it's much much much more likely that there was some kind of cross contamination.
Surely the only point of the maths you've used is to show that the maths is pointless :)I recognise your reasoning - at least in view of the assumptions I made in my earlier number crunching - where I tried to illustrate the unaffected chances of one person randomly receiving a false positive, which was turn followed by another etc, etc.
Firstly, let me dispel the idea that “cross contamination” could be a likely cause of this string of false positives on LFTs.
I don’t know if you’ve ever taken one but they are a rather private affair. You unwrap the swab, swirl round each of your nostrils, rub the swab around the acidity regulating liquid in the test tube, leave for a minute or two and then pour four drops on to the indicating tablet.
You then wait a maximum of 15 minutes and you have your result, but if you have a positive - as I’ve sadly experienced regularly this week (who would have guessed I’m currently self-isolating) the red lines appears almost straight away against T letter in the presence of a relative high level of the virus.
The process of taking an LFT is not a social one!
Other players could not have cross contaminated them unless (preposterous I know) they had intended to.
The wildly unlikely odds I outlined earlier were on the assumption that one incidence wouldn’t affect the incidence of another. They would not if the tests could be shown to be working and not part of faulty batch.
However, given that the chances of having that 15 false positive tests occurring randomly in succession is less than the number of stars in the universe squared, then either the supplier of the tests was as fault, and it was faulty batch or Liverpool’s staff did something out of the ordinary. That is: did something daft, without realising for instance. A mess up we might learn from, should we have been given the chance to do so that is by Mr Parry at the EFL.
I would proffer the best course action is for those that lost out in missing the Arsenal game, the fans that had booked travel and accommodation, the tv companies and advertisers that missed out to know submit a joint (class action) claim to recover damages simultaneously against both Liverpool AND the test providers.
One of the two parties is guilty and the courts won’t mind who pays out to the victims. They will be instructed to sort it out between themselves. That will bring resolution to this.
Surely the only point of the maths you've used is to show that the maths is pointless :)
Clearly these weren't 15 independent events, so you may as well just ignore that option. I know many on here think that Klopp personally asked whoever definitely did have Covid to spit all over the rest of the tests, but most likely it was a dodgy batch or, as you say, someone did something stupid (or maybe, and I don't know what the timeline is, but given they'll all have been likely triple vacced, and many of them have already had it, they got an incredibly weak infection that clearer up within days - my wife tested negative, then positive then negative with covid over the course of a few days recently).
We also don't know the protocol for the tests the PL players take, but given they've been taking tests for many months, they would either be very good at it, or quite possibly have slacked off over time. Maybe they still have someone helping them take them, or someone collecting them all together. While all these events are uncommon, they're all possible, and given the number of sports teams taking regular tests, may well have been statistically almost inevitable.
I work in an office and with scousers.I know a few of them. We are only separated by 30 miles down the EastLancs. There is good and bad in everyone. We just have banter. They ain't bad lads. Work in construction you meet them all the time. You kids who work in an office in town would not know. I have more Scouse kids on my watsapp than Manchester lads. They are just the same as us. The only difference between us and them is the accent.
On the rare occasion when one is found guilty a royal pardon is issued by the home SecretaryThey never get investigated for anything. Take the bus incident, nobody arrested. The spitting incident earlier this season, they looked into it , nothing. And don’t get me started with the hacking of our computer systems. Teflon scousers.