Due to inaccuracies in drug testing procedures(e.g.,false positives and false ne

Question

Due to inaccuracies in drug testing procedures(e.g.,false positives and false negatives),in the medical field the results of a drug test represent only one factor in a physician's diagnosis .Apply Bayes' Rule for making inferences.Use the following example.In a population of 2,000 patient ,suppose 200 are illegally using Marijuana .Of the users,suppose 100 would test positive for Marijuana .Of the nonusers,suppose 18 would test positive.

a-Given the patient is a user ,find the probability that a drug test for Marijuana will yield a positive result.This Probability represents the sensitivity of the drug test.

b-Given the patient is a non user ,find the probability that a drug test for Marijuana will yield a negative result.(This probability represents the specificity of the drug test).

c-If a patient tests positive for a drug ,use Bayes' Rule to find the probability that the patient is really using Marijuana .(This Probability represents the positive predictive value of the drug test).

Explanation / Answer

given total Population N=2000

Peoples using marijuana (A) =200

Peoples not using marijuana(B) =1800

let E is event that test showing +ve

N is event that test showing -ve result

now

number of peoples test shows +ve and in actual using marjiuana=n(A and E) =100

number of peoples test shows +ve and in actual not using marjiuana=n(B and E)=18

so

P(A) =0.1 P(B)=0.9

P(A and E) =100/2000 =0.05 P(B and E) =18/2000 =0.009

so

P(A and N) =100/2000 =0.05 P(B and N) =1782/2000 =0.891

a)

we have to find P(E | A)=?

so

P(E|A) =P(E and A)/P(A) =0.05/0.1 =0.5

b)

we have to find P(N|B)=?

so

P(N|B) =P(N and B)/P(B) =0.891/0.9 =0.99

c)

we have to find P(A|E)

P(A|E) =P(A and E)/P(E)

=0.05/0.05+0.009

=0.05/0.059

=0.847

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