Question

*5.98 A biometric security device using fingerprints erroneously refuses to admit 1 in 1,000 authorized persons from a facility containing classified information. The device will erroneously admit 1 in 1,000,000 unauthorized persons. Assume that 95 percent of those who seek access are authorized. If the alarm goes off and a person is refused admission, what is the probability that the person was really authorized?

Answer 1

Given that 95% of persons are granted access, we know that probability of alarm going off is 5% or 0.05

Also, when the alarm goes off probability of it being an authorized person is 1/ 1000 = 0.001

Therefore, joint probability of alarm going off and person being an authorized person is

P = 0.05 x 0.001 = 0.00005 or 5 in 10,000.

Answer 2

By using Bayes’ theorem        

Let “A” be the event for an individual was authorized and “B” be the event for an individual was refused entry so,

The device will erroneously admit 1 in 1,000,000 unauthorized persons,

                                                   

95 % percent of those who seek access is authorized. P(A) = 0.95

We have to calculate P(A/B) by using the above formula.

P(B) = (0.95) () + (0.05) () = 0.05094995

P(A/B) =

Answer 3

By using Bayes’ theorem         

Let “A” be the event for an individual was authorized and “B” be the event for an individual was refused entry so, 

The device will erroneously admit 1 in 1,000,000 unauthorized persons,

                                                   

95 % percent of those who seek access is authorized. P(A) = 0.95

We have to calculate P(A/B) by using the above formula.

P(B) = (0.95) () + (0.05) () = 0.05094995

P(A/B) =