Question

Suppose that the probability of a person having a headache is 0.01, that the probability of a person having a fever given that the person has a headache is 0.4, and that the probability of a person having a fever is 0.02. Find the probability that a person has a headache given that the person has a fever.

Answer 1

Let $A$ denote the event "a person has a headache", $B$ denote the event "a person has a fever".

Given $P(A)=0.01, P(B|A)=0.4, P(B)=0.02.$

Then

$$P(B|A)=\dfrac{P(A\cap B)}{P(A)}, P(A|B)=\dfrac{P(A\cap B)}{P(B)}$$

$$P(A|B)=\dfrac{P(B|A)P(A)}{P(B)}=\dfrac{0.4(0.01)}{0.02}=0.2$$

The probability that a person has a headache given that the person has a fever is 0.2.