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Let $P(Disease)$ and $P(\neg Disease)$ are the probabilities of a patient having a particular disease or not. If $P(+)$ and $P(-)$ are the probabilities of a test detecting this disease or not.

Is the probability of a test being wrong $P(+/\neg Disease) + P(-/Disease)$, $P(\neg Disease/+) + P(Disease/-)$ or $P(+ \cap \neg Disease) + P(-\cap Disease)$

I think the first one is right, but cannot explain why the other two are wrong.

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No, the last one is right.

As an example, assume that the test never gives the correct outcome, so $P(+ | \lnot \mathrm{Disease}) = 1$ and $P( - | \mathrm{Disease}) = 1$.

The first one of the options then is equal to $2$, which means it cannot be correct, since a probability should always lie in the interval $[0, 1]$.

The last one is correct because it takes into account both the accuracy of the test, as well as how many people actually have the disease. If a test has $P(+ | \mathrm{Disease}) = 0.3$, and $P( - |\lnot \mathrm{Disease}) = 1.0$, but only $0.0001\%$ of the population has the disease, then the test will give the correct result for $99.99997\%$ of the population.

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