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here's where prevalence comes in...
You've nailed where the confusion emerged.
If you change the prevalence amongst the population you're altering the odds for your own infection status
Yeah - that makes perfect sense. And where the confusion emerged. The odds of a result being one thing or another can be dependent on prevalence (or any other number of factors, actually) independent of the test's claimed/stated/tested accuracy. I agree with this, but, as I think is clear now, I initially made the mistaken assumption that you were making a bolder claim than you were. This is my fault.
But maybe we should argue about something else now?
(I'm going for a bike ride).
Indeed.
But, for fear of overdoing it, here's where prevalence comes in...
If I know I'm negative then taking the test I've described above I know that the result will be correct 95% of the time.
If I know I'm positive then taking the test I've described above I know that the result will be correct 95% of the time.
But people don't know whether they are negative or positive, hence the need to take the test. This is where prevalence comes in. Let's say it's 10% amongst the population (to make my numbers easy again). [EDIT] Changed from 5% to make it obvious this is nothing to do with the test sensitivity/specificity.
Also I'm only going to do the calculations for a positive test result.
So there's a 90% chance I'm truly negative. If I'm truly negative then the chance of a false positive result is 5%. This route gives me a 4.5% chance of getting a positive result.
Conversely there's a 10% chance I'm truly positive. If I'm truly positive then the chance of a true positive result is 95%. This route gives a 9.5% chance of getting a positive result.
So a positive result in a test (given these assumptions) gives me 67.9% chance (9.5% / (9.5%+4.5%)) of getting an accurate result given the balance between truly positive and false positive. (That's not accuracy for the test overall. If you do the same calculations for a negative outcome it's 99.7% or similar.)
Your status when you take the test is unknown, so you have to use the general prevalence to answer the question of "How accurate is a specific outcome?"
If you change the prevalence amongst the population you're altering the odds for your own infection status, which affects the derived accuracies for the two test outcomes. It's not so much that the numbers/status of other people that are infected affects your test, but it's the prevalence that needs to be taken into account for your possible initial infection status.