A pharmaceutical company manufacturing a virus detection kit wants to determine the probability of a person without the virus receiving a positive text result.It is estimated that the probability of infected potential users of the kit is 10%.According to the company laboratory test results,1 out of 100 viral-free people tested positive (false positive).On the other hand,1 out of 200 infected people tested negative (false negative).A person has just used the detection kit manufactured by the company and the results showed they had the virus.What is the probability that the person is viral-free?

Question

Quizplus · Accepted Answer

We need to use Bayes' theorem to solve this problem. Let A be the event of a person being viral-free and B be the event of testing positive for the virus. We want to find P(A|B), the probability of being viral-free given a positive test result. 
Using Bayes' theorem: 
P(A|B) = P(B|A) * P(A) / P(B) 
where P(B|A) is the probability of testing positive given that the person is viral-free, P(A) is the probability of a person being viral-free (complement of infected potential users, i.e., 90%), and P(B) is the probability of testing positive (which can be calculated using the false positive and false negative rates). 
P(B|A) = 0.01 (from the given information) 
P(A) = 0.9 
P(B) = P(B|A) * P(A) + P(B|A') * P(A') = 0.01 * 0.9 + 0.99 * 0.1 * 0.9 = 0.1071 
(where P(A') is the probability of being infected) 
Plugging in the values: 
P(A|B) = 0.01 * 0.9 / 0.1071 = 0.083 or 8.3%
Therefore, the answer is C.

A Pharmaceutical Company Manufacturing a Virus Detection Kit Wants to Determine