A drug test for athletes has a 5% false positive rate and a 10% false negative rate. Of the athletes tested, 4% have actually been using the

Question

A drug test for athletes has a 5% false positive rate and a 10% false negative rate. Of the athletes tested, 4% have actually been using the prohibited drug. If a randomly chosen athlete tests positive, what is the probability that the prohibited drug has been used

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Faith 1 week 2021-09-13T02:43:53+00:00 1 Answer 0

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    2021-09-13T02:45:32+00:00

    Answer:

    0.4286 or 42.86%

    Step-by-step explanation:

    The probability that an athlete has used drugs and tests positive is given by the probability that he had been using drugs (4%) multiplied by the probability of a true positive (1 – false negative):

    P(D\ and\ +) = 0.04*(1-0.10)=0.036

    The probability that an athlete has NOT used drugs and tests positive is given by the probability that he had not been using drugs (100% – 4%) multiplied by the probability of a false positive (5%):

    P(N\ and\ +) = (1-0.04)*0.05=0.048

    Finally,  the probability that the prohibited drug has been used given that an athlete tested positive is:

    P(D|+)=\frac{P(D\ and\ +)}{P(D\ and\ +) \ +\ P(N\ and\ +)} \\P(D|+)=\frac{0.036}{0.036+0.048}=0.4286

    The probability is 0.4286 or 42.86%.

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