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’?

 ◦ 0.4 ◦ 0.5 ◦ 0.6 ◦ 0.64 ◦ 0.8

I suppose the answer is 0.8 since that is the probability that a result is accurate.

Answer 1

If test shows "infected", then it is positive

If test shows "not infected", then it is negative

P(positive/infected)=0.80

P(negative/not infected)=0.80

P(positive/not infected)=1-0.80=0.20

P(infected)=0.20

P(not infected)=1-0.20=0.80

So,

P(positive)=P(positive/infected) $\times$ P(infected)+P(positive/not infected) $\times$ P(not infected)

$=0.80 \times 0.20+0.20 \times 0.80 \\ =0.32$

Using Bayes rule:

$P(infected/positive)=P(positive/infected) \times \frac{P(infected)}{P(positive)} \\ =0.80 \times \frac{0.20}{0.32} \\ =0.5$

The answer is 0.5