# What is the probability that this person really has the disease?

A test for a certain disease has a probability $\frac{4}{5}$ of detecting the disease when it ispresent and a probability of $\frac{1}{10}$ of falsely \detecting" it when it is not present.

The proportion of people afflicted by the disease for a given population is 15%.

If a person from the population is randomly selected and gives a positive result to the test, what is the probability that this person really has the disease? (Answer: 24/41).

I'm struggling to get that answer. I honestly just don't know how to approach this question.

• Try applying these fractions/percentages to an actual population (say, 200 people). – Semiclassical Aug 16 '14 at 16:59

Let $D$ and $\neg D$ represent the events that the patient has and does not have the disease respectively, and let $T$ and $\neg T$ represent the event that the test is positive or negative respectively.
$P(D \mid T)$