The value that the doctors SHOULD have been calculating is the PPV (Positive Predictive Value) (See: https://en.wikipedia.org/wiki/Positive_and_negative_predictive_values)
This is a probability that given a positive test result that the test is a True Positive.
This is calculated as:
PPV = sensitivity x prevalence / (sensitivity x prevalence + (1-specificity) x(1-prevalence))
In the sample question, you don't really have enough information to calculate this exactly, as they only give the "False Positive Rate", this allows the calculation of specificity (1 - FPR) as 0.95. The sensitivity is not defined, lets just assume for the sake of the example that the sensitivity (1 - FNR) is also 0.95.
So....
PPV = (0.95 x 0.001) / (0.95 x 0.001 + (1 - 0.95) x (1 - 0.0001))
PPV = 0.018664
i.e. Given the prevalence of the disease, the likelihood that a positive test result represents a TRUE positive is: 1.8% Even if you assume that the sensitivity of the test is 99.5%, the PPV still comes out around 16%. It's a bit of a silly question, as the data required to actually calculate the answer isn't supplied (you need the False Negative Rate as well as False Positive Rate to do the calculations properly. Tests rarely include this required information though).
A test with a sensitivity as low as 95% should only ever be used as a screening test, never as a diagnostic test. Basically, you want something that is very sensitive, but with the possibility of a high False Positive Rate which is used as a screening test, combined with a follow up test that has a much higher specificity (very low False Positive Rate). Only when the confirmatory test comes back positive can you make any conclusions about the diagnosis (even then, you should be doing the PPV calculation to determine the likelihood that the positive result is a True positive.
I was running a rough calculation of PPV during the main panic stages of COVID, when they were doing mass testing and reporting the results daily. During most of it, the PPV, as per my calculations (which did have some assumptions, as the specific accuracy details of the tests were not disclosed and the prevalence had to be somewhat inferred by the number of positive test results), was running around 10%. So, when they were saying that they tested 50k people, and found 75 "cases", probably 90% of those "cases" were false positives.
I'm not a doctor, not even very good at math.
The question is concerning a person who "has a positive result", so it seems like the 1/1000 is irrelevant information.
Based on your answer, my initial thinking must not be correct.
If 100 out of 1000 cavemen have an apple, but 5% of the apples are actually a plastic replica of an actual apple.. then only 95 cavemen have apples grown from a tree.
Using your formula, for the above statment to determine how many cavemen have actual apples, I get a value of .65519?
Thank you. The question throws out "prevalence of 1/1000" but then asks about only people with a positive result... a result that is wrong 5% of the time. It doesn't ask about the odds of some random dude having the disease, which is what people are trying to answer. It only asks about people who already have tested positive. And if the test gives a false positive 5% of the time, then 5% of the people who tested positive got the wrong result. People who tested negative aren't part of the actual question. It's more about reading comprehension than maths skills.
If thats an accurate take, then there is a subset of people who are lacking in one life skill, who believe they are more intelligent than the majority because they don't get the same answer..
Now I think I've got to re-examine/reaffirm my beliefs on covid, the 6 million, and lord knows what else..
In that example, you're not calculating how many cavemen have apples. Rather you are calculating the probability, given that you are that a caveman has an apple, that is a real apple rather than a fake one.
So given a prevalence of 0.1 specificity of the test of 0.95 then a PPV around 0.65 seems about right. If your "test" detected an apple, there's about a 65% chance that it's really an apple.
I guess I'm too retarded to wrap my head around 5/100 equating a 65% chance. I knew there was a reason I chose to stay out of professions that puts the lives of others in my hands. :D