Maths question

[PecanPiez]

I looked to see whether there’s a section of mumset specifically about maths, and there’s not, so I’m posting here as the question relates to Coronavirus.

Apparently lateral flow Covid tests have a very low rate of false positives, but the false negative rate is up to 50%: amp.theguardian.com/world/2021/apr/15/what-is-lateral-flow-covid-test-how-accurate

If the false negative rate is 50%, and someone does 3 lateral flow tests in the space of a few minutes, and they are all negative, what percentage chance is there that that result is correct?

Thanks in advance to any maths pros out there!

The science/numbers and graphs long running thread can probably help. It's been a brilliant source of calm, quiet, logical fact during the whole pandemic.

0.125

@MaxNormal has a maths degree I seem to remember them saying.

No its not

If each test has a 50% chance of being a false negative, then 3 tests also have a chance of being false negatives

Nannyandjohn may have mentioned their maths degree a couple of times

They are mutually exclusive events. So your chance of a false positive will be 50 percent every time you do it. No matter how many times you do it.

It's like when you have a child, the chance of your child being a girl or boy are the same however many times you have had a child.

The question you are asking is what are the chances of having a boy?

This can be calculated (I can't, like you be arsed to google) but it's not a useful number.

Therefore the answer to this is get a PCR

False negatives will generally be caused by low viral load and/or poor swabbing technique - if one's a false negative chances are that all tests done at the same time will be.

You can't work this on the maths basis of probability.

One of the big reasons for false negatives is poor technique, and a person who has poor technique might always return a false negative.

The overall probability of a false negative being 50% doesn't mean that each positive individual taking an individual test has a probability of 50% of a false negative.

They may have a low viral load (LFTs aren't very sensitive) or they may not be doing the test properly (far less likely to get a positive if you are administering the test yourself) or they may be a child (not sure of the data for using them on children).

It depends if the error is random or systematic. Most of it is probably systematic - not doing it properly.

Each test is not affected by the result of thr previous test, so the probability is the same as any other outcome. An example is that tossing a coin, HHH, TTT, HTH, THT, TTH, THH, HHT, HTT all have the same probability.

I really don't know enough about the workings and failings of lateral flow tests to know if each result would be truly independent. I do know enough about them to know that they are designed for asymptomatic testing, and that if you have symtoms a PCR test is recommended.

The same chance with each individual test.

But it’s not random, if you are positive surely you would be more likely to get a positive with more tests?

@Nootkah

Each test is not affected by the result of thr previous test, so the probability is the same as any other outcome. An example is that tossing a coin, HHH, TTT, HTH, THT, TTH, THH, HHT, HTT all have the same probability.

Except if the negative is due to poor technique, then the results are NOT independent of each other.

If you have symptoms you need to leave the lateral flows alone and get a PCR. If you don't then there is no point in doing more than one lateral flow because you will probably keep getting the same result.

@noblegiraffe

I remember you answering this question well! Sorry!!

@noblegiraffe

Oh sorry you have again !

If the false negative rate is 50%, and someone does 3 lateral flow tests in the space of a few minutes, and they are all negative, what percentage chance is there that that result is correct?

The answer to this is not fixed as it depends on how likely the person was to have covid when they took the test, which varies depending on location, time, reason they took the test, etc.

It depends on the reason for false negative results; test manufacturing error can be excluded by multiple tests (from different packs), bad technique might be slightly mitigated by trying 3 times and too little virus to be detectable won't be any different each time. Overall, you may as well use the provided statistic of 50%. That isn't the correct statistic though, the worst case is 42% (www.gavi.org/vaccineswork/how-accurate-are-lateral-flow-tests).

If you've had symptoms or been near a covid positive person, that 42% isn't enough to be reassured that you are safe to be near vulnerable people (or any large crowds). If you're just checking as part of a big herd of people to reduce asymptomatic spread, then it's literally much better than nothing.