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Solution

a)

EXACTLY THREE:

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 =    7      
p = the probability of a success =    0.45      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.291847746 [ANSWER]

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AT LEAST THREE:

P(at least 3) = 1 - P(0) - P(1) - P(2)

Using the same formula above,

P(0) = 0.015224352
P(1) = 0.087194018
P(2) = 0.21402168

Thus,

P(at least 3) = 1 - 0.015224352 - 0.087194018 - 0.21402168 = 0.68355995 [ANSWER]

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AT MOST THREE:

P(at most 3) = P(0) + P(1) + P(2) + P(3)

Using the same formula above,

P(3) = 0.291847746

Thus,

P(at most 3) = P(0) + P(1) + P(2) + P(3)

P(at most 3) = 0.015224352 + 0.087194018 + 0.21402168 + 0.291847746

= 0.608287796 [ANSWER]

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B)

P(between 2 and 4) = P(2) + P(3) + P(4)

Using the same formula,

P(4) = 0.23878452

Thus,

P(between 2 and 4) = P(2) + P(3) + P(4)

P(between 2 and 4) = 0.21402168 + 0.291847746 + 0.23878452

= 0.744653946 [ANSWER]

P(between 2 and 4) = 0.21402168 + 0.291847746

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