Question

One half percent of the population has a particular disease. A test is developed for the disease. The test gives a false positive $3\%$ of the time and a false negative $2\%$ of the time. What is the probability that Amit (a random person) tests positive?

• A. $0.476$
• B. $0.035$
• C. $0.523$
• D. $0.232$

Answer 1

Answer: (B)

Let $D$ be the event that Amit has the disease. Let $T$ be the event that Amit's test comes back positive. We are told that $P(D)=0.005$, since $1/2\%$ of the population has the disease. We also have $P(T|D)=.98$, since $2\%$ of the time a person having the disease is missed ("false negative"). We are told that $P(T|D^c)=.03$, since there are $3\%$ false positive.
Required probability :
$P(T)=P(T|D)P(D)+P(T|D^c)P(D^c)$
$=(.98)(.005)+(.03)(.995)=0.035$