Another way to illustrate the base rate fallacy is to get rid of any randomness for a moment and imagine a perfectly regular and deterministic toy universe: Fo

Another way to illustrate the base rate fallacy is to get rid of any randomness for a moment and imagine a perfectly regular and deterministic toy universe: For the disease test, suppose every patient has a unique ID, starting at #0 and counting up consecutively. Because it's a toy universe, we can say that the actually infected patients are exactly the ones with IDs #0, #10000, #20000 etc. The test will come back positive if either the patient is infected (i.e. false-negative rate is 0, P(positive|infected) is 1.0) or if the ID is one of #1, #1001, #2001, #3001, etc. This results in a base rate of 1/10000 and a false positive rate of (almost) 1/1000. If you now look at all the IDs for which the test will be positive, those will be: #0, #1, #1001, #2001, ..., #9001, #10000, #10001, #11001, #12001, ..., #20000, #20001, #21001, etc etc. It's easy to see that for each true positive (IDs that end with 0), there are 10 false positives (IDs that end with 1) - so the probability that some specific ID from that list is a true positive P(infected|positive) is 1/10 = 0.1, i.e. it's more likely the patient is not infected than that they are. (It's still more likely that they are infected than it would be for a random pick from the general population: 1/10 vs 1/10000) Now consider a second population with a higher base rate: Now the infected patients are #0, #50, #100, #150 etc, i.e. the base rate is 1/50. If you look at the IDs with positive tests again, you get #0, #1, #50, #100, #150, ..., #1000, #1001, #1050, ..., #2000, #2001 etc etc. Now there are 50 true positives for each false positive, i.e. P(infected|positive) is something like 50/51 =~ 0.98. So the test suddenly got much more "reliable", even though the false-positive rate didn't change, only through a change in base rate.