Probablity

Question

A given medical test has a 90% chance of detecting a disease if it is present, and a 6% chance of detecting the disease if it is not present. Given that currently 5% of the population have the disease, what is the probability that if the test says you have the disease, you really have the disease?

Answer 1

Let's start by making a tree of possible outcomes. (multiply probability of events; +Disease ∩ +test = 0.05*0.9)

 

 

0.1 (-Test) -->  0.005 False negative

0.05 (Disease)

0.9 (+Test) -->  0.045 True positive

 

 

0.06 (+Test) --> 0.057 False positive

0.95 (No Disease)

0.94 (-Test) -- > 0.893 True negative

 

 

So then what is the probability that if the test says you have the disease, you really have the disease?

 

P{Disease|+test} = P{True positive}/P{+Test}

 

The chance of a True positive is 0.045 (calculated above) but there are two ways to test positive, as either a true positive or a false positive. P{+Test} = 0.045+0.057 = 0.102

 

P{Disease|+Test} = 0.045/ 0.102

P{Disease|+Test} = 0.44