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.
(a). What is the probability that Joe (a random person) tests positive?
(b). Joe just got the bad news that the test came back positive; what is the probability that Joe has the disease?

Answer 1

Answer

a) 0.035

b) 0.14

Step-by-step explanation:

Let J be the event that Joe has the disease.

Let The be the event that Joe's test is positive.

Pr(J) = 1/2%

= 0.5/100 = 0.005

Pr(J') = 99.5%

= 0.995

Pr(T|J) = 98%

= 0.98 since 2% if the time if a person having the disease is omitted ("false negative ")

Pr(T|J') = 3% = 0.03 since there are 3 false positives

a( Pr(T) = us the probability that Joe tests positive

Pr(T) = Pr(T|J)* P(J) + Pr(T|J')*Pr(J')

= (0.98*0.005) + (0.03*0.995)

= 0.00049 + 0.02985

= 0.03475

= 0.035

b) Pr( J|T) = Pr(JnT) / Pr(T)

= (Pr(T|J)*Pr(J)) / Pr(T)

= (0.005*0.98) / 0.035

= 0.0049/0.035

= 0.14