Question

Patients of a clinic are tested for a particular desease. For each patient, the result of the test – ‘infected’/’not infected’ – is correct with the probability 0.8. Suppose that 20% of the patients are infected. What is the probability that a given patient is indeed infected if his/her test result shows ‘infected’?

Answer 1

Solution:

Notations:

CT: correct test report

In: Infected

$P(CT|In )=0.8, P(CT|In')=0.2$

$P(In)=0.2, P(In')=0.8$

$P(In| CT)=\dfrac{P(In)\times P(CT|In)}{P(In)\times P(CT|In)+P(In')\times P(CT|In')}$

$=\dfrac{0.2\times 0.8}{0.2\times 0.8+0.8\times 0.2} \\=\dfrac{0.16}{0.32}=\dfrac12$