I know how to apply Bayes Theorem to determine the likelihood a patient has a given disease if multiple trials return a positive result. In this case, if two trials both return positive:
$$ P(\text{User}\mid++) = \frac{P(\text{User})P(+\mid\text{User})^2}{P(\text{User})P(+\mid\text{User})^2+(1 − P(\text{User})) P(+\mid\neg\text{User})^2} $$
However, how could I modify the above formula if one conducts three trials, and exactly two tests come out positive? What would be the likelihood that the patient has the disease then?
I would assume that $_3C_2$ comes into play as well, but I'm lost as to how to incorporate the negative result into the probability calculation.