There's a really interesting mathematical 'paradox' with false positives along the same line as the birthday paradox. It's an important one also that many doctors get wrong. Let's say there is a terrible disease that 0.01% (1 in 10,000) of the population is infected with, and we develop a test that is 99% accurate. You go to the doctor and get a test. Oh no, it's positive. What are the chances that it's a false positive?
Intuition would tell you 1% since the test is 99% accurate. Intuition, as it often is, is wrong. Take a sample size of 'x', we'll use 10,000. How many people will be infected? 1. How many false positives will there be? 1% * 10,000 = 100. So there will be 100 false positives, and 1 real positive. So it turns out there's a greater than 99% chance that your result is a false positive, even though the test is genuinely 99% accurate.
Here's a fun one. So we decide to run the test twice. And AGAIN it comes up positive. What are the odds that it's a false positive? Again, the exact same logic holds - there are 101 people we are testing. There will be about 1 false positive and about 1 real infection. So the odds of it being a false positive once again are about 50%.
Intuition would tell you 1% since the test is 99% accurate. Intuition, as it often is, is wrong. Take a sample size of 'x', we'll use 10,000. How many people will be infected? 1. How many false positives will there be? 1% * 10,000 = 100. So there will be 100 false positives, and 1 real positive. So it turns out there's a greater than 99% chance that your result is a false positive, even though the test is genuinely 99% accurate.
Here's a fun one. So we decide to run the test twice. And AGAIN it comes up positive. What are the odds that it's a false positive? Again, the exact same logic holds - there are 101 people we are testing. There will be about 1 false positive and about 1 real infection. So the odds of it being a false positive once again are about 50%.