Question

A genetic test is used to determine if people have a predisposition for thrombosis, which is the formation of a blood clot inside a blood vessel that obstructs the flow of blood through the circulatory system. It is believed that 3% of people actually have this predisposition. The genetic test is 99% accurate if a person has the predisposition and 98% accurate if a person does not have the predisposition. What is the probability that a randomly selected person who tests positive for the predisposition actually has the predisposition? (In other words, what is the probability of a true positive result?)

Answer 1

0.6049

Explanation:

Let's define the following events for a randomly selected person

HP: the person has the predisposition

NP: the person does not have the predisposition

TP: the person tests positive

TN: the person tests negative

We know that P(HP) = 0.03 (because 3% of people actually have the predisposition), so, P(NP) = 0.97,

P(TP | HP) = 0.99 (the genetic test is 99% accurate if a person has the predisposition),

P(TN | NP) = 0.98 (the genetic test is 99% accurate if a person does not have the predisposition),

P(TP | NP) = 0.02. We are looking for the probability that a randomly selected person who tests positive for the predisposition actually has the predisposition, i.e., P(HP | TP). By Bayes' formula

P(HP | TP) = (P(TP | HP)P(HP))/(P(TP | HP)P(HP) + P(TP | NP)P(NP)) = [(0.99)(0.03)]/[(0.99)(0.03) + (0.02)(0.97)] = 0.6049