What is the probability that a random person selected is diseased, given a positive detection? There is a method to detect certain disease with 99% sensitivity and 90% specificity. And also it is known that 5% of the population has this disease. What is the probability a random person selected from the population actually has this disease given that the method detects so?

Sensitivity is the probability that someone that has the disease tests positive for it, and specificity is the probability that someone who does not have the disease tests negative. See here for a more detailed explanation. So if we let $X$ represent a positive test and $D$ represent the patient having the disease then $P(X|D)=0.99$ and $P(X|\
eg D)=0.1$. Apply Bayes rule and take two asprin if symptoms persist.