37 of adults say cashews are their favorites kind of nuts y

37% of adults say cashews are their favorites kind of nuts . you randomly select 12 adults and ask each to name his/her favorite nuts. find the probability that the number who say cashews are their favorite nuts is

(A) exactly 3 (b) at least 4, and (c) at most two. if convenient , use technology to find the probability

(A) P(3)= round to the nearest thousands as needed

(B) P(x >_ 4)= round to the nearest thousdandth as needed

(C) P(x <_ 2)= round to the nearest thousandth as needed.

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 =    0.37      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.174217909 [answer]

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 =    0.37      
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   ) =    0.29466996
          
Thus, the probability of at least   4   successes is  
          
P(at least   4   ) =    0.70533004 [answer]

c)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.37      
x = the maximum number of successes =    2      
          
Then the cumulative probability is          
          
P(at most   2   ) =    0.12045205 [answer]