Periodically, the Federal Trade Commission (FTC) monitions the pricing accuracy

Question

Periodically, the Federal Trade Commission (FTC) monitions the pricing accuracy of electronic checkout scanners at stores to ensure consumers are charged the correct price at checkout. A pat study of over 100,000 items found that one every 30 items is priced incorrectly by the scanners. Suppose the FTC randomly selects 45 items at a retail store and check the accuracy of the scanner price at each. Find the following probabilities with and without the Poisson approximation to the binominal distribution. What is the probability that exactly one of the 45 items is priced incorrectly by the scanner? What is the probability that at most two of the 45 items is priced incorrectly by the scanner?

Explanation / Answer

a)

1 defect for every 30 items means 1*(45/30) = 1.5 defects for every 45 items.

Note that the probability of x successes is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    1.5      
          
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.33469524 [ANSWER]

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b)

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    1.5      
          
x = the maximum number of successes =    2      
          
Then the cumulative probability is          
          
P(at most   2   ) =    0.808846831 [ANSWER]

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