The company policy for customer service reps gives time off for positive reviews

Question

The company policy for customer service reps gives time off for positive reviews. If in the first 20 calls a customer service rep handles in a day, 13 or more elect to take a subsequent survey and rate the service rep as ' excellent', then the company gives the rep his or her final hour of work that day off, paid. Ellie receives excellent reviews from about 30% of the calls she handles. Assuming she always receives at least 20 calls in the first 7 hours of a workday, on what percentage of her 8-hour workdas does Ellie get the final hour off? Please show how you got the answer.

Explanation / Answer

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.3      
x = our critical value of successes =    13      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   12   ) =    0.99872112
          
Thus, the probability of at least   13   successes is  
          
P(at least   13   ) =    0.00127888 = 0.1279% [ANSWER]

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