Question

Tests for use of any particular drug have false positive rates (how often a non-user will test positive) and false negative rates (how often someone who uses the drug will test negative); the false negative rates are typically higher than the false positive rates, when the tests are conducted correctly. Suppose a test with a 5% false positive rate and a 15% false negative rate is being used to screen a population where 1% of people are actually using the drug being tested for. Assume the test is being conducted correctly. Express all probabilities as decimals, rounded to four places.

1. If someone is selected at random from the population, what is the probability they would test positive for the drug? (Note: this is equivalent to asking what proportion of the population would test positive for the drug.)

2. If someone tests negative, what is the probability they are using the drug?

3. If someone tests positive, what is the probability they are not using the drug?

Answer 1

Here,the terms proportion and probability are used interchangeably. So,

Probability of people using drug = 0.01

Probability of people not using drug = 1 - 0.01 = 0.99

false negative means when people were using drug but tested nagative

probability of testing negative, when people are using drug = 0.15

So, probability of testing positive, when people are using drug = 1 - 0.15 = 0.85

false positive means testing positive when people not actaully using drug

probability of testing positive when people are not using drug = 0.05

So, probability of testing negative, when people are not using drug = 1 - 0.05 = 0.95

Now, using the law of conditional probability , P(B/A) = P(A and B)/P(A)   (B/A means B given A)

P (A and B) = P(A) * P(B)

1. P (using drug and tests positive) = P(using drug) * P(tests positive given using drug) = 0.01 * 0.85 = 0.085

Similiarly, P (not using drug and tests positive) = 0.99 * 0.05 = 0.0495

P (has a positive test result) = 0.085 + 0.0495 =  0.1345

2. In same way,

P (using drug and tests negative) = P(using drug) * P(tests negative given using drug) = 0.99 * 0.15 = 0.1485

Similiarly, P (not using drug and tests negative) = 0.01 * 0.95 = 0.095

P (has a negative test result) = 0.1485 + 0.095 =  0.2435

P (using drug giiven that test negative) = P (using drug and testing negative) / P(test negative) = 0.1485 / 0.2435 = 0.6099

3. P (not using drug giiven that test positive) = P (not using drug and testing positive) / P(test positive) = 0.0495 / 0.1345 = 0.3680

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