sting Problem: Suppose a drug test is 99% sensitive and 98% specific That is, th

Question

sting Problem: Suppose a drug test is 99% sensitive and 98% specific That is, the test will produce 99% true positive results for drug users and 98% true negative results for non-drug users eare users of the drug. We need to find the solution to the question: al tests positive (+), what is the probability he or she is a User (U) Suppose that 0.5% of peopl f a randomly selected indi Bayes' rule: p( B) (a) Write down Bayes' rule needed to answer the above question in terms of +,U,U. (b) In the case of the above testing problem write down the expression for p(B) in terms of a summation. (c) Identify the prior in the testing problem by writing down its algebraic expression. (d) If a randomly selected individual tests positive (+), what is the probability he or she is a User (U)? (e) If a randomly selected individual tests positive (+), what is the probability he or she is not a User (U)

Explanation / Answer

(a)

In this case we need to find the solution to the following equation:

P(U | +) = P(U AND +)/P(+) = P(U)*P(+ | U)/P(+)

(b)

In this case we have the following summation equation:

P(+) = P(+ | U)*P(U) + P(+ | U')*P(U')

(c)

Data given to us is as follows:

P(U) = 0.005

So, P(U') = 0.995

Also, P(+ | U) = 0.99 and P(- | U') = 0.98

Now,

P(+ | U) = P(+ AND U)/P(U)

So,

P(+ AND U) = P(U)*P(+ | U) = 0.005*0.99 = 0.00495

We also have the following equation:

P(+ | U') + P(- | U') = 1

So,

P(+ | U') = 1-0.98 = 0.02

So,

P(+) = P(+ | U)*P(U) + P(+ | U')*P(U') = 0.99*0.005 + 0.02*0.995 = 0.02485

(d)

Here we need to calculate P(U | +)

P(U | +) = P(U)*P(+ | U)/P(+) = (0.005*0.99)/0.02485 = 0.199

(e)

Here we need to calculate P(U' | +)

Using equation:

P(U | +) + P(U' | +) = 1

So,

P(U' | +) = 1-0.199 = 0.801

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