Question

A medical test gives a correct result 90% of the time for infected individuals and 95% of the time for non-infected individuals. All members of the population are tested. The probability that a person is infected given they have a positive test result is 0.6.

What proportion of the population is infected?

Answer 1

From the given data, the following Table is calculated:

Let p = proportion of the population that is infected

	Infected	Not infected	Total
Test positive	0.90 p	0.05 - 0.05 p	0.05 + 0.85 p
Test negative	0.10 p	0.95 (1 - p) = 0.95 - 0.95 p	0.95 - 0.85 p
Total	p	1 - p	1.00

Given:
P(Infected/ Test positive) = 0.6

By Multiplication Theorem of Probability, we have:

P(Infected/ Test positive) = P(Infected AND Test positive) / P(Test Positive)

Substituting values, we get:

So, we get:

0.6 (0.05 + 0.85 p) = 0.90 p

i.e.,

0.03 + 0.51 p = 0.90 p

So,

0.39 p = 0.03

So,

p = 0.03/0.39

= 0.0769

So,

Answer is:

0.0769