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 are 10:19. This means that Pr[disease|+++] = 10/29, just over 1/3.
You may have obtained 1/2 by a double confusion between odds and probabilities. If your prior had been Pr[disease]=1/21, then we'd have prior odds of 1:20 and posterior odds of 1:2 (which is a probability of 1/3, not of 1/2).
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