$30$% of the country has a disorder. A blood test has developed that has a $90$% true positive rate(the probability that a person will test positive given that they have the disorder) and a $25$% false positive rate (the probability that a person will test positive given that they don't have the disorder). Every person who has tested positive uses medicine which has a $20$% rate of causing acne. What's the probability that someone has acne and actually have the disorder.
I know how to find probably of people who are actually ill given a positive rate. But I don't know how to use the acne rate to find the probability of ill people.
$P(D)$ = $30$% the probability of having the disorder
$P(\bar{D})$ = $70$% probability of being healthy
If E stands for having acne and T+ for testing positive:
$P(E) = P(E|D,T+)\cdot P(D, T+) + P(E|\bar{D}, T+)\cdot P(\bar{D}, T+)$,
Of course I agree with @Henno Brandsma but if this is the case, it is not necessary to know what's the rate of false positive (EDIT: they do not have the disorder thus they do not affect $D \cap A$)
thus the solution is simply
$P(T^+ \cap D)=0.3\times 0.9=0.27$
and $P(D\cap A)=0.27\times0.2=5.4\%$