A patient is thought to have one of three diseases $A_1$, $A_2$ and $A_3$ whose probabilities under the given conditions are $1/2$, $1/6$ and $1/3$, respectively. A test is carried out to help the diagnosis and it yields a positive result with a probability of $0.1$ for disease $A_1$, a probability of $0.2$ for disease $A_2$ and a probability of $0.9$ for disease $A_3$. A(nother) test is conducted 5 times and the results are positive 4 times and negative once. What is the probability of each disease after testing?
Let $P(P_1) = 0.1$. I can't decide whether $P(P_1)$ is $P(P_1|A_1)$. A positive result for $A_1$ does not necessarily mean the person has disease $A_1$, the results could be incorrect. Furthermore, I am not sure what I am supposed to find here: the probability of each disease after testing?? Is it the probability of each disease given that the results are positive or negative? Or a total probability? Should I use Bayes' rule here?