25 of college students say they use credit cards because of

25% of college students say they use credit cards because of the rewards program. You randomly select 10 college students and ask eacgh to name the reason he or she uses credit cards. Find the probability that the number of college students who say they use credit cards because of the rewards program is (a) exactly 2, (b) more than 2, and (c) between 2 and 5 inclusive. round answers to the nearest thousands.

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 =    0.25      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.281567574 [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 =    10      
p = the probability of a success =    0.25      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   2   ) =    0.525592804
          
Thus, the probability of at least   3   successes is  
          
P(more than   2   ) =    0.474407196 [ANSWER]

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c)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    2      
x2 =    5      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    10      
p = the probability of a success =    0.25      
          
Then          
          
P(at most    1   ) =    0.24402523
P(at most    5   ) =    0.980272293
          
Thus,          
          
P(between x1 and x2) =    0.736247063   [ANSWER]

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