Question

Tuberculosis (TB) is a disease caused by bacteria that are spread through the air from person to person. If not treated properly, TB disease can be fatal. People infected with TB bacteria who are not sick may still need treatment to prevent TB disease from developing in the future.

It is estimated that about 3% of the population in a given country is infected with TB bacteria. There is a skin test for TB infection. However, the test is not always accurate. The probability that someone who is infected with TB bacteria will test positive is 0.99. The probability that someone who is not infected with TB bacteria will test negative is also 0.99.

Suppose that a randomly chosen person takes the skin test, and the outcome of the test is positive. What is the probability that this person is infected with TB bacteria?

Answer 1

Given,

P(TB) = 0.03

P(Positive | TB) = 0.99

P(Negative | ~TB) = 0.99

Now,

P(~TB) = 1 - P(TB) = 1 - 0.03 = 0.97

P(Positive | ~TB) = 1 - P(Negative | ~TB) = 1 - 0.99 = 0.01

By law of total probability,

P(Positive) = P(TB) P(Positive | TB) + P(~TB) P(Positive | ~TB)

= 0.03 * 0.99 + 0.97 * 0.01

= 0.0394

Given, the outcome of the test is positive, probability that this person is infected with TB bacteria

= P(TB | Positive)

= P(Positive | TB) P(TB) / P(Positive) (By Bayes theorem)

= 0.99 * 0.03 / 0.0394

= 0.7538071