Question

At the O’Hare International Airport (ORD) travelers have to walk through a metal detector to pass through to the departure area. The security team knows from prior experience that 8% of all of travelers who walk through it have metal on their person. If someone who passes through does have metal on them, the metal detector will go off 92% of the time. If they do not have metal on them, it will only go off 4% of the time. You are the TSA agent manning the station, and watch a traveler about to pass through the metal detector.

1. What is the prior probability that the traveler has metal on their person?
2. If this traveler walks through the metal detector and the alarm sounds, what is the probability that they have metal on them? (Hint: Use a tree diagram to solve this problem.)
3. Consider the Denver International Airport (DIA), where 3.2% of all of the individuals who walk through it have metal on their person. Assume the accuracy of the metal detector at DIA is the same as that of ORD when someone does and doesn’t have metal on them. Suppose the DIA metal detector sounds an alarm for a traveler. (You may assume that this traveler is a randomly drawn person from the population of travelers at DIA.) Do you expect the probability that they have metal on them to be lower, higher, or the same as that of the ORD traveler for whom the alarm sounded? Why?
4. Suppose that after the alarm sounds on the ORD traveler the first time, they refute the result and say they would like to pass through the metal detector once more. What is the new prior probability you should use on this second pass for them having metal on their person?
5. The alarm sounds again on this second pass. Should the new probability of them having metal on their person be lower, higher, or the same as what you calculated in part (2)?

Answer 1