Periodically, the Federal Trade Commission (FTC) monitors the pricing accuracy o

Question

Periodically, the Federal Trade Commission (FTC) monitors the pricing accuracy of electronic checkout scanners at stores to ensure consumers are charged the correct price at checkout. A past study of over 100,000 items found that one of 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 binomial distribution. (a) What is the probability that exactly one of the 45 items is priced incorrectly by the scanner? (b) What is the probability that at most two of the 45 items is priced incorrectly by the scanner?

Explanation / Answer

a)

WITHOUT POISSON:

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    45      
p = the probability of a success = 1/30 =   0.033333333      
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.337495479 [ANSWER]

WITH POISSON:

Here, the mean number of successes = n p = 45*(1/30) = 1.5.

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]

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

b)

WITHOUT POISSON:

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    45      
p = the probability of a success =    0.033333333      
x = the maximum number of successes =    2      
          
Then the cumulative probability is          
          
P(at most   2   ) =    0.811023617 [ANSWER]

WITH POISSON:

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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