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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?

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  1. 25 July, 06:12
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    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
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