Question

Problem 7.
A certain virus infects five in every 100 people. A test used to detect the virus in a person is
positive 70% of the time if the person has the virus, and 9% of the time if the person does not
have the virus. Using the Bayes’s theorem, if a person tests positive, determine the probability
that the person is infected.

Answer 1

Let A be the event that person is infected.

and B be the event that the person tests positive, then

Probability of person being infected, P(A) = 5/100 = 0.05

Probability of person not being infected = 0.95

Now to find probability of B, the person can test positive by two ways - having infection and not having infection

Probability of testing positive when person is infected = 0.05 * 0.7 = 0.035 (0.7 comes from 70% given in question)

Probability of testing positive when person is not infected = 0.95 * 0.09 = 0.0855 (0.09 comes from 9% given in question)

Total probability of testing positive, P(B) = 0.0855 + 0.035 = 0.1205

Also, P(B/A) = 0.7

We have to calculate P(A/B).

Using Bayes Theorem,

P (A/B) = P(B/A) * P(A) / P(B)

A/B means A given B has ocurred

So, P(A/B) = 0.7 * 0.05 / 0.1205 = 0.2904