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  4. 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?

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Q. 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?

Probability - Part 2

Solution:

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