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)

$0.035$

Answer 2

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 : $P(T) = P(T|D) P(D) + P(T|D^c) P(D^c)$ $= (.98) (.005) + (.03) (.995) = 0.035$