Bayes theorem and a test detecting a disease 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.

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.
