I don't understand how an LFT can detect coronavirus in someone with no symptoms, but if someone is unwell it is not going to detect coronavirus in them. Explain it to me, please! I want to understand.
I don't have symptoms of Coronavirus anyway.
It's sort of complicated, and not quite as black and white as that, but I'll have a go (caveat: I may be wrong!).
There are two aspects to the question - one is the mechanics and efficacy of the test itself, the other is mathematical and about statistical sampling and how it affects results.
The first is easier to talk through: The way the LFT works is by detecting coronavirus proteins picked up on the swab. It needs a certain amount of that protein to be able to detect - too little and it just won't show up.
The amount of protein picked up on the swab depends on a number of factors: the amount of viral protein in the subject (ie viral load); the amount of protein in the subject in the location the swabs are taken (the primary locus of the infection moves around, and is different in different people); the skill of the person in being able to sample the proteins.
The first one of these is
generally going to be correlated with symptomatic status - but it's not a perfect correlation - it's entirely possible for someone with symptoms to have a lower viral load than someone without. This on its own can lead to the situation you are asking about.
The second one adds more uncertainty to the first - for instance someone with severe lower lung infection might not actually have much virus left in their upper airways (different immune responses affect the upper airways), so they might be really quite ill, but not have enough protein in the nose/throat to be able to detect it. Similarly with heavy gut load infection, or other patterns of infection (this is quite correlated with the time in the course of the disease the sample is taken - early infections tend to be more in the upper airway, and it migrates down into the rest of the body if the infection is not checked).
The third aspect then adds more uncertainly again - even if an unwell person has high viral load in the upper airways, they might just not be very good at swabbing (it's not easy to do right without having been shown). This is a systematic effect - ie someone who's not very good at it will probably be not very good at it every time they try, so continually fail to pick up a positive result.
Hope that all makes some sense - I will have a go at the conditional probability part of the question later on when I've had some time to remind myself how it works