33 of adults say that cashews are their favorite kind of nut

33% of adults say that cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name his or her favorite nut. Find the number probablity that the number who say cashews are their favorite nuts is (a) exactly 3 (b) at least 4 and (c) at most 2.

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 =    12      
p = the probability of a success =    0.33      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.21509867 [answer]

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b)

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 =    12      
p = the probability of a success =    0.33      
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.402659566
          
Thus, the probability of at least   4   successes is  
          
P(at least   4   ) =    0.597340434 [answer]

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c)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.33      
x = the maximum number of successes =    2      
          
Then the cumulative probability is          
          
P(at most   2   ) =    0.187560896 [answer]