A fingerprint matching system uses an algorithm (T) to \"match\" fingerprints to

Question

A fingerprint matching system uses an algorithm (T) to "match" fingerprints together under comparison. The system can come up with a "positive match" T+ or a "negative match" T-. The fingerprints under comparison can truly be from the same person M+ or not M-. The matching system has the following performance characteristics:

The true positive rate of the matcher is: Pr(T+ | M+) = 0.95

The false positive rate of the matcher is: Pr(T+ | M-) = 0.003

The probability that any two randomly selected fingerprints from different people in the human population will "match" is conservatively estimated by experts as: Pr(M-) = 0.000001

. What is the probability the matcher will come up with a "positive match" on any given comparison, Pr(T+)?  

b. What is the true negative rate of the matcher?  

c. Compute Pr(M+|T+):  

d. Compute Pr(M-|T+)  

e. Compute Pr(M-|T-)

Explanation / Answer

a)

By Bayes' Rule,

P(T+) = P(M+) P(T+|M+) + P(M-) P(T+|M-)

= (1-0.000001)*0.95 + 0.000001*0.003

= 0.949999053 [ANSWER]

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b)

Here,

P(T-|M-) = 1 - P(T+|M-) = 1 - 0.003 = 0.997 [ANSWER]

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c)

Here,

P(M+|T+) = P(M+) P(T+|M+)/P(T+) = (1-0.000001)*0.95/0.949999053 = 0.999999997 [ANSWER]

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d)

As

P(M-|T+) = 1 - P(M+|T+) = 1 - 0.999999997 = 3.1579*10^-9 [ANSWER]

***********************

e)

As

P(M-|T-) = P(M-) P(T-|M-)/P(T-)

Then

P(M-|T-) = 0.000001*0.997/(1-.949999053) = 1.99396E-05 [ANSWER]

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