A market researcher working for a retail store found that 80

A market researcher working for a retail store found that 80 percent of randomly selected  
customers are repeat customers. 10 customers are randomly selected. What is the probability that

a) Three of the customers selected are repeat customers?

b) Between five and seven (inclusive) of the selected customers are repeat customers?

c) If you select 20 customers, how many customers would you expect to be repeat customers?

Solution

a)

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 =    10      
p = the probability of a success =    0.8      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.000786432 [answer]

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b)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    5      
x2 =    7      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    10      
p = the probability of a success =    0.8      
          
Then          
          
P(at most    4   ) =    0.006369382
P(at most    7   ) =    0.322200474
          
Thus,          
          
P(between x1 and x2) =    0.315831091   [answer]

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c)

If n = 20, then

E(x) = n p = 20*0.8 = 16 [ANSWER]