Question

Ronald has developed a 10-second test for strep throat. Wanting to capitalize on his discovery, he decides to open up a drive-up McClinic on which a customer's throat is swabbed by a nurse at one window and antibiotics are dispersed at a second window of the strep test is positive. The strep test returns a positive result when a customer really does not have strep throat ( a "false positive") with the probability p, 0 < p < 1. The strep test returns a negative result when a customer really has strep throat (a "false negative") with probability q, 0 < q < 1. If the probability that a customer driving to Ronald's McClinic has strep throat is r, 0 < r < 1, find the probability that a customer who drives away from the McClinic with antibiotics really does have strep throat.

Answer 1

X : A customer driving to Ronald's McClinic has strep throat

: Not X : A customer driving to Ronald's McClinic does not have a strep throat

Probability that a customer driving to Ronald's McClinic has strep throat is r i.e.

P(X) = r

P() = 1-P(X) = 1-r

P : The strep test returns a positive

The strep test returns a positive result when a customer really does not have strep throat ( a "false positive") with the probability p i.e

P(P|) = p

Q(Not P ; ) : The strep test returns a negative

The strep test returns a negative result when a customer really has strep throat (a "false negative") with probability q

P(Q|X) = P( |X) = q

P(P|X) = 1-P(|X) = 1-q

Probability that a customer who drives away from the McClinic with antibiotics really does have strep throat = P(X|P)

By Bayes Theorem,

P(X) P(P|X) = r(1-q)

the probability that a customer who drives away from the McClinic with antibiotics really does have strep throat :