Assume that X is a binomial random variable with n 10 and p

Assume that X is a binomial random variable with n = 10 and p = 0.80. 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 =    10      
p = the probability of a success =    0.8      
x = the number of successes =    9      
          
Thus, the probability is          
          
P (    9   ) =    0.268435456 [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 =    10      
p = the probability of a success =    0.8      
x = the number of successes =    8      
          
Thus, the probability is          
          
P (    8   ) =    0.301989888 [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 =    10      
p = the probability of a success =    0.8      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   7   ) =    0.322200474
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.677799526 [answer]