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

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