You are working in a primary care office Flu season is start

You are working in a primary care office. Flu season is starting. For the sake of public health, it is critical to diagnose people with the flu. The prior probability that someone who is walking through your door has the flu is 0.1. If someone has the flu, their probability of having a runny nose is 0.99. However, if someone doesn’t have the flu, e.g. just has the cold (they will (think that they) have something, otherwise they wouldn’t seek out your office), their probability of having a runny nose is 0.9. Someone comes in and has a runny nose. What is the probability that this person has the flu? A. 0.5 B. 1 C. 0.1 D. 0 E. 0.11

Solution

Let

F = has flu
R = has runny nose

Thus,

P(F|R) = P(F) P(R|F) / P(R)

As, by Bayes\' Rule,

P(R) = P(F) P(R|F) + P(F\') P(R|F\') = 0.1*0.99 + (1-0.1)*0.9 = 0.909

Thus,

P(R) = 0.1*0.99/0.909 = 0.108910891 [ANSWER]