Assume that X is a binomial random variable with n 16 and p

Assume that X is a binomial random variable with n = 16 and p = 0.66. Calculate the following probabilities. (Round your intermediate and final answers to 4 decimal places.)

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 =    16      
p = the probability of a success =    0.66      
x = the number of successes =    15      
          
Thus, the probability is          
          
P (    15   ) =    0.010684592 = 0.0107 [answer]

b)

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 =    16      
p = the probability of a success =    0.66      
x = the number of successes =    14      
          
Thus, the probability is          
          
P (    14   ) =    0.041281377 = 0.0413 [answer]

c)

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 =    16      
p = the probability of a success =    0.66      
x = our critical value of successes =    14      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   13   ) =    0.946737739
          
Thus, the probability of at least   14   successes is  
          
P(at least   14   ) =    0.053262261 = 0.0533 [answer]