Assume that X is a binomial random variable with n 21 and p

Assume that X is a binomial random variable with n = 21 and p = 0.86. 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 =    21      
p = the probability of a success =    0.86      
x = the number of successes =    20      
          
Thus, the probability is          
          
P (    20   ) =    0.143984703 [ANSWER]

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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 =    21      
p = the probability of a success =    0.86      
x = the number of successes =    19      
          
Thus, the probability is          
          
P (    19   ) =    0.234393703 [ANSWER]

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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 =    21      
p = the probability of a success =    0.86      
x = our critical value of successes =    19      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   18   ) =    0.57950362
          
Thus, the probability of at least   19   successes is  
          
P(at least   19   ) =    0.42049638 [ANSWER]