A tourist bureau survey showed that 75 of those who seek inf

A tourist bureau survey showed that 75% of those who seek information about the state actually come to visit. The office received 8 requests for information. Find:

a) the probability that more than five of the people will visit

b) the probability that at least 2 of the people will NOT visit

c) the probability that less than 4 of the people will visit

d) the probability that 4-6, inclusive, of the people will visit

e) what is the expected number of visitors from this group

Solution

a)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    8      
p = the probability of a success =    0.75      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   5   ) =    0.321456909
          
Thus, the probability of at least   6   successes is  
          
P(more than   5   ) =    0.678543091 [ANSWER]

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b)

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 =    8      
p = the probability of a success =    0.25      
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   ) =    0.367080688
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.632919312 [ANSWER]

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c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    8      
p = the probability of a success =    0.75      
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.027297974
          
Which is also          
          
P(fewer than   4   ) =    0.027297974 [ANSWER]

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d)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    4      
x2 =    6      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    8      
p = the probability of a success =    0.75      
          
Then          
          
P(at most    3   ) =    0.027297974
P(at most    6   ) =    0.632919312
          
Thus,          
          
P(between x1 and x2) =    0.605621338   [ANSWER]

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e)

E(x) = n p = 8*0.75 = 6 [ANSWER]