SOLUTION: A single die is rolled 22 times. What is the probability that a six is rolled exactly once, if it is known that at least one six is rolled?

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Question 1126505:
A single die is rolled 22 times.
What is the probability that a six is rolled exactly once, if it is known that at least one six is rolled?
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
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            Let us solve the problem step by step. 


1.  The probability that a six is rolled exactly once, if it is known that at least one six is rolled, is equal to the ratio,


    where the numerator is the probability that a six is rolled exactly once,

    while the denominator is the probability that at least one six is rolled.



2.  The numerator is equal to  C%5B22%5D%5E1%2A%281%2F6%29%2A%285%2F6%29%5E%2822-1%29 = 22%2A%281%2F6%29%2A%285%2F6%29%5E21.



3.  The denominator is the complement to the probability that no one six is rolled.


    The probability that no one six is rolled is  %285%2F6%29%5E22;  hence, the complement to it is  1 - %285%2F6%29%5E22.



4.  Therefore, the probability under the question is 

        %2822%2A%281%2F6%29%2A%285%2F6%29%5E21%29%2F%281+-+%285%2F6%29%5E22%29 = 0.081172 = 8.1172%  (approximately).     ANSWER

Solved.