Suppose a lab test has the following statistics for detecting a certain disease.
$A$ is the event that the test result is positive, and $B$ is the event the tested person has the disease.
$P(A \mid B) = 0.95$ and $P(A \mid B') = 0.002,$ and $0.5$% of population actually has the disease.
Find the probability that a person has the disease given that the test result is positive.
The probability looks like $P(D \mid A),$ but how do I find the probability of $D$??
$B$ is the event that the tested person has the disease. So, if we assume that this person was just a random person from the population as a whole (in other words, if we know nothing more about this person other than that it is some member of the population at large), then $P(B)$ is the same as the probability of anyone from the population having the disease, i.e. $0.5$%