A laboratory blood test is 99% effective in detecting a certain disease when its infection is present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Explanation

Let `E_(1) and E_(2)` be the respective events that a person has a disease and a person has no disease.
Since `E_(1) and E_(2)` are events complimentary to each other, we have
`P(E_(2))=1-P(E_(1))=1-0.001=0.999`
Let A be the event that the blood test result is positive.
`P(A//E_(1))` =P (result is positive given the person has disease) `=99%=0.99`
`P(A//E_(1))=P` (result is positive given that the person has no disease) `=0.5%=0.005.`
Probability that a person has a disease, given that his test result is positive, is given by `P(E_(1)|A).`
By using Bayes' theorem, we obtain
`P(E_(1)//A)=(P(E_(1)).P(A//E_(1)))/(P(E_(1)).P(A\\E_(1))+P(E_(2)).P(A\\E_(2)))`
`=(0.001xx0.99)/(0.001xx0.99+0.999xx0.005)`
`=(0.00099)/(0.005985)=22/133`