It is amazing how difficult this is to explain this to people! So I have come up with two simple diagrams.

The first is a simple example:

If this one sinks in, proceed to an example with the latest statistics:

Just a note about where the numbers come from.
The cases and tests come from the latest 7-day averaged ONS daily figures here:
https://coronavirus.data.gov.uk/

The estimated percentage of the population infected is here:
https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/conditionsanddiseases/datasets/coronaviruscovid19infectionsurveydata
(Specifically, cell B38 on sheet 1a of their spreadsheet)

What I am assuming here is that the tested group represents the general population. This becomes increasingly true as test number increase.

If the ONS estimate of infections is correct, it would entail a very low specificity (93.5%) that has dropped from 99% at the beginning of September. I have already commented on this, suggesting that the tripling of number of tests at that time has led to poor quality control in the laboratories and/or using too many cycles.

Here is one for the conspiracy theorists. Should there be those who wish to prolong the anti-covid measures, this would be one way to do it while absolving themselves from any blame. That is, put pressure on the laboratories to cause the testing to become inaccurate and over-predict the 'case' numbers.

But I digress ...