*This is garbage - please DO NOT post about things you have no understanding about! You're treating them like tossing a fucking coin.
If one test doesn't register a +ve, the next test will give exactly the same result, unless the first test failed due to user error.*
@strangeshapedpotato, the poster's working is right (though they have taken an optimistic view of the false negative rate).
The LFT's false negative rate depends on a) viral load, b) tester (unsurprisingly, test done by Porton Down scientists is more often correct - ie the same as a PCR would give - than one supervised by someone trained via video) and c) device. Minimum false negative rate (best device, high viral load, Porton Down) is around 25%. Maximum I have seen published (standard device, students supervised by trained volunteers) = 97%.
Each test's result is independent of the previous test (in the sense that if test A was a false negative, the probability that test b will be falsely negative is independent of this). The only caveat is that, in a developing infection, viral load is likely top increase and thus the false negative rate may go down a little.
Using 50% as a reasonable 'best case scenario' for false negatives:
If you are actually positive, but take 4 LFTs, your probability of 4 false negatives is, at best, 1/16 (2 to the power of 4 being 16) or 6.25%.
It is likely to be higher, because the false negative rate in normal 'unsupervised' use is likely to be higher. The Birmingham university study would put it at 88.5%, for example.
However, combined with that is the probability of being positive. I seem to remember that the latest ONS figure was about 1 in 560, so at a population level your probability of 4 false negatives while actually (randomly) positive = 0.01%.
The 1 in 560, though, is population level. If you are in Blackburn, and work in a school which has outbreaks, for example, then the 1 in 560 will grossly understate your probability of being positive.