Question

A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.78. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.02. It is estimated that 19 % of the population who take this test have the disease.
If the test administered to an individual is positive, what is the probability that the person actually has the disease?

Answer 1

Solution:

Let’s start by writing down the information we know from reading the problem :
P(Positive|Has Disease) = 0.78
P(Positive|No Disease) = 0.02
P(Has Disease) = 0.19
P(No Disease) =1-P(Has Disease) = 1- 0.19 = 0.81
What we want to find is P(Has Disease|Positive), and we can find it using Bayes Theorem :

P(Has Disease|Positive) = P(Positive|Has Disease) P(Has Disease) / P(Positive)

We already have the probabilities in the numerator, but we need to calcu-late the probability that a test is positive. We can do this by combining theknowledge we have already:

P(Positive) =P(Positive∩Has Disease) +P(Positive∩No Disease)
P(Positive) =P(Positive|Has Disease)P(Has Disease)+P(Positive|No Disease)P(No Disease)P(Positive)
= (0.78)(0.19) + (0.02)(0.81) = 0.1644
Now let’s plug this in to our equation :
P(Has Disease|Positive) =P(Positive|Has Disease)P(Has Disease)/P(Positive)
P(Has Disease|Positive) = (0.78)(0.19)/ 0.1644 = 0.9015