Please show all work to justify final answer and answer both

Please show all work to justify final answer and answer both parts. Thank you.

A particular basketball player has a free-throw percentage of 92%. Assume they take 20 free throws in practice. What is the probability of making exactly 15? Of making more than 18?

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 =    20      
p = the probability of a success =    0.92      
x = the number of successes =    15      
          
Thus, the probability is          
          
P (    15   ) =    0.014544912 [answer]

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 =    20      
p = the probability of a success =    0.92      
x = our critical value of successes =    18      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   18   ) =    0.483144359
          
Thus, the probability of at least   19   successes is  
          
P(more than   18   ) =    0.516855641 [ANSWER]