Question

It is estimated that about 3% of 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

Solution:- If randomly chosen person takes the skin test, and the outcome of the test is positive, then the probability that this person is infected with TB bacteria is 0.7538.

P(Infected) = 0.03, P(Not infected) = 1 - 0.03 = 0.97

$P(\frac{Positive}{Infected}) = 0.99$

$P(\frac{Negative}{Not.Infected}) = 0.99$, $P(\frac{Positive}{Not.Infected}) = 1-0.99=0.01$

$P(\frac{Infected}{Positive}) =\frac{P(Infected)\cdot P(\frac{Positive}{Infected})}{P(Infected)\cdot P(\frac{Positive}{Infected})+P(Not.Infected)\cdot P(\frac{Positive}{Not.Infected})}$

$P(\frac{Infected}{Positive}) =\frac{0.03\cdot 0.99}{0.03\cdot 0.99+0.97\cdot 0.01}$

$P(\frac{Infected}{Positive}) =0.7538$