# 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.

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.