Air traffic controllers monitor the flights of aircrauft and help to ensure safe

Question

Air traffic controllers monitor the flights of aircrauft and help to ensure safe takeoffs and landings. In the US air traffic controllers are required to undergo periodic random drug testing. A urine test is used an initial screening test due to its low cost and ease of implementation. One such urine test, the Triage Panelfor Drugs of Abuse plus TCA, detects the presence of drugs and has a sensitivity of 0.96 and a specificity of 0.93. Based on past drug testing of air traffic controllers, the FAA reports that the probability of drug use is 0.007. Use this information to answer the next 2 questions. 13) What is the probability that an air traffic controller returns a positive drug test? a) 0.0067 b) 0.007 c) 0.0695 d) 0.0762 e) None of the above are close 14) Given that an air traffic controller returns a positive drug test, what is the probability that they have been using drugs? a) 0.007 b) 0.088 c) 0.912 d) 0.96 e) None of the above are close

Explanation / Answer

Let D shows the event that person uses drug so

P(D) =0.007

P(Dc) = 1 - P(D) = 1-0.007 = 0.993

And we have

P(P|D) = 0.96

P(N|Dc) = 0.93

By the complement rule we have

P(N|D) = 1 - P(P|D) = 0.04

P(P|Dc) = 1 - P(N|Dc) = 0.07

13:

By the law of total probability , the test gives positive result is

P(P) = P(P|Dc)P(Dc) + P(P|D)P(D) = 0.07 *0.993 + 0.96 *0.007 = 0.07623

Hence, option D is correct.

14:

P(D|P) = [P(P|D)P(D) ] / P(P) = [0.96 * 0.007] / 0.07623 = 0.0882

Hence, option (b) is correct.

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