Question

You are designing a new system for detecting an explosive device. Your current design has a 1% false negative rate and a 5% false positive rate. The estimated frequency of actual devices through a typical security check point where the system will be employed is 1 in 5,000. Use the variables E to represent the explosive device and D (or D1 and D2) to represent the detector(s).

a)What is the probability that when the detector indicates the presence of an explosive device that the person is actually carrying one? Show your work?

b)On average, how many people will have to be detained and inspected for every device detected?

c)Now suppose you have additional equipment that gives you an independent test for the same types of devices, but it’s noisier: the false negative rate is 5% and the false positive rate is 10%. What is the probability of a device when both of them signal a detection? Show your work.

Answer 1

Solution

Back-up Concepts

In terms of a test,

true positive => test is positive when the patient has the disease ...................................................... (1)

true negative => test is negative when the patient does not have the disease.................................... (2)

false positive => test is positive when the patient does not have the disease .................................... (3)

false negative => test is negative when the patient has the disease ...................................................... (4)

Now, to work out the solution,

Current design has a 1% false negative rate => vide (4)

P(the person has the device given that the detector indicates absence of device) = 0.01........................................ (5)

Current design has 5% false positive rate => vide (3)

P(the person does not have the device given that the detector indicates presence = 0.05....................................... (6)

Part (a)

Probability that when the detector indicates the presence of an explosive device that the person is actually carrying one

= 1- P(the detector indicates the absence of an explosive device when the person is actually carrying one)

= 1 – 0.05 [vide (6)]

= 0.95 Answer

Part (b)

Given, ‘The estimated frequency of actual devices through a typical security check point where the system will be employed is 1 in 5,000.’ =>

On average, number of people to be detained and inspected for every device detected = 5000 Answer 2

Part (c)

New design has 10% false positive rate => vide (3)

P(the person does not have the device given that the detector indicates presence = 0.10 .................................. (7)

Now, given the two detectors are independent,

probability of a device when both of them signal a detection

= 1- probability of no device when both of them signal a detection

= 1- (probability of no device when D1signals a detection x probability of no device when D2 signals a detection)

= 1 – (0.05 x 0.1) [vide (6) and (7)]

= 0.995 Answer 3

DONE