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Did I use Bayes' formula correctly here?
Prior: 1/20
12/20 chance that test A returns correctly +
16/20 chance that test B returns correctly +
12.5/20 chance that test C returns correctly +
Odds of correct diagnosis?
I got 1/2
Let's assume that every test has the same probability of returning the correct result, regardless of what it is (e.g., if + is correct, then Pr[A returns +] = 12/20, and if - is correct, then Pr[A returns +] = 8/20).
The key statistic for each test is the ratio Pr[X is positive|disease] : Pr[X is positive|healthy]. This ratio is 3:2 for test A, 4:1 for test B, and 5:3 for test C. If we assume independence, we can multiply these together, getting a ratio of 10:1.
If your prior is Pr[disease]=1/20, then Pr[disease] : Pr[healthy] = 1:19, so your posterior odds ... (read more)