Calculate the following probabilities for the following bino

Calculate the following probabilities for the following binomial distributaions.

(a) P(X=4) if n=12 and p=0.42

(b) P(x<2) if n=11 and p=0.37

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 =    0.42      
x = the number of successes =    4      
          
Thus, the probability is          
          
P (    4   ) =    0.197254223 [ANSWER]

*****************

b)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    11      
p = the probability of a success =    0.37      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.046291724
          
Which is also          
          
P(fewer than   2   ) =    0.046291724 [ANSWER]