A survey found that 80 of US household do some online bankin

A survey found that 80% of US household do some online banking. In a collection of 220 US households, what’s the probability that:

a. Exactly 175 do some online banking?

b. Less than 175 do some online banking?

c. 175 or more do some online banking?

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 =    220      
p = the probability of a success =    0.8      
x = the number of successes =    175      
          
Thus, the probability is          
          
P (    175   ) =    0.065616832 [ANSWER]

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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 =    220      
p = the probability of a success =    0.8      
x = our critical value of successes =    175      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   174   ) =    0.394154814
          
Which is also          
          
P(fewer than   175   ) =    0.394154814 [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 =    220      
p = the probability of a success =    0.8      
x = our critical value of successes =    175      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   174   ) =    0.394154814
          
Thus, the probability of at least   175   successes is  
          
P(at least   175   ) =    0.605845186 [ANSWER]