Question 1176923
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The probability that at least one of the 5 vaccinated children will develop the infection is the COMPLEMENT 

to the probability that NO ONE will develop the infection. 


So the answer to the problem's question is  P = 1 - {{{(1-0.05)^5}}} = 1 - {{{0.95^5}}} = 0.2262 = 22.62%.    <U>ANSWER</U>
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Solved.