Consider the following scenario. Some US bank notes, n = 100, are seized on a su

Question

Consider the following scenario. Some US bank notes, n = 100, are seized on a suspect. The probability that a randomly selected bank note from the general population contains cocaine is about 57%. The probability that a bank note contains cocaine if it has been involved in drug trafficking is about 83%.

a. What is the probability that more than 50 notes contain cocaine if they were not involved in drug trafficking?  

b. What is the probability that more than 50 notes contain cocaine if they were involved in drug trafficking?  

c. In 2015 there were about 38.1 billion US bank notes in circulation. How many on average are contaminated with cocaine?  

d. What is the standard deviation in the number of US bank notes that contained cocaine in 2015?  

e. What is the probability that a bank note in your wallet contains cocaine?  

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

a. 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)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    100      
p = the probability of a success =    0.57      
x = our critical value of successes =    50      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   50   ) =    0.095035538
          
Thus, the probability of at least   51   successes is  
          
P(more than   50   ) =    0.904964462 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    100      
p = the probability of a success =    0.83      
x = our critical value of successes =    50      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   50   ) =    3.77156E-14
          
Thus, the probability of at least   51   successes is  
          
P(more than   50   ) =    1 [ANSWER]

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c)

As n = 38.1 billion, p = 0.57,

u = mean = np =    21717000000 [ANSWER]

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d)

Also,

s = standard deviation = sqrt(np(1-p)) =    96634.93157 [ANSWER]

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e)

I am not involved in drug trafficking. Hence,

P(cocaine) = 0.57 [ANSWER]

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