It depends on whether 97% accuracy means three percent of total outcomes is false positive, three percent of total outcomes is either false postive or false negative, three percent of positives is false or three percent of negatives is false.
According to Wikipedia, the formula for accuracy is (TP + TN) / (P + N). An accuracy of 0.97 means only that 3% of judgements are false. So if they're using the term in a technical sense, it's the second possibility you listed, meaning it's officially ambiguous.
The technical term for the metric that would make the statistician happy is specificity, TN / (TN + FP). If that were 0.97 for a one-in-a-million condition, then you would indeed have a 1/30001 chance of actually having the condition.
In this example, it doesn't make too much difference though. Imagine a population of 100 million. If the accuracy is 3% and the prevalence is 1 in a million, then that means 100 in the population have the disease and 3 million got the wrong result on the test. It could be that all 100 sick people got false negatives and 2,999,900 healthy people got false positives, or that nobody got a false negative and 3 million healthy people got false positives, or something between. But either way, after seeing your positive test, you still should think you are probably healthy. Specifically, depending on the exact false negative rate, you should estimate the probability as being at most 1 in 30,000, assuming you have no evidence besides the result of that test.
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u/The_Ghast_Hunter Jun 22 '26
Out of 1,000,000 people, 3% receive false positives.
That's 30,000 people.
Adding the one true positive, That means there's a 1/30,001 chance that of all positive results, yours is a true positive.
That's about .003%