There is a standard question in probability about the probability of a person having a disease given a positive test. I am interested in an extension of this: What is the probability two people are negative given both tests are negative? What if we assume the two are either both positive or negative - that it cannot be the case one person is positive and the other is not.
2026-03-29 23:27:17.1774826837
Extension of Bayes' Rule when looking at more than one person for diagnostic testing knowing both either have or don't have the disease.
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If both patients have the same disease state, $D$, then a plausible assumption is that the two test results, $T_1,T_2$, are conditionally independent for a given disease state.$$\mathsf P(T_1{=}s,T_2{=}t,D{=}d)=\mathsf P(T_1{=}s\mid D{=}d)~\mathsf P(T_2{=}t\mid D{=}d)~\mathsf P(D{=}d)$$ You should be given the reliability and accuracy of the test, and the population rate for the disease state, so use this to find: $\mathsf P(D{=}0\mid T_1{=}0,T_2{=}0)$, the probability for patients not having the disease when given both test results are negative.