Question

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. Why is the probability that if six randomly selected telephone numbers are called, at least three of them will be busy?

Solution

Here n = 6
$straight p space equals space 1 over 15 comma space space space straight q space equals space 1 minus 1 over 15 space equals space 14 over 15$
P(at least three numbers are busy) = P(3) + P(4) + P(5) + P(6)
$equals space straight C presuperscript 6 subscript 13 space open parentheses 1 over 15 close parentheses cubed space open parentheses 14 over 15 close parentheses cubed space plus space straight C presuperscript 6 subscript 4 space open parentheses 1 over 15 close parentheses to the power of 4 space open parentheses 14 over 15 close parentheses squared space plus space straight C presuperscript 6 subscript 5 space open parentheses 1 over 15 close parentheses to the power of 5 space open parentheses 14 over 15 close parentheses to the power of 1 space plus space straight C presuperscript 6 subscript 6 space open parentheses 1 over 15 close parentheses to the power of 6 equals space fraction numerator 6 space cross times space 5 space cross times space 4 over denominator 1 space cross times space 2 space cross times 3 end fraction cross times space open parentheses 1 over 15 close parentheses cubed space cross times fraction numerator open parentheses 14 close parentheses cubed over denominator left parenthesis 15 right parenthesis cubed end fraction plus fraction numerator 6 space cross times space 5 over denominator 1 space cross times 2 end fraction space cross times space open parentheses 1 over 15 close parentheses to the power of 4 space cross times space fraction numerator open parentheses 14 close parentheses squared over denominator left parenthesis 15 right parenthesis squared end fraction plus 6 over 1 cross times open parentheses 1 over 15 close parentheses squared space cross times 4 over 15 plus 1 space cross times space open parentheses 1 over 15 close parentheses to the power of 6 equals space open parentheses 1 over 15 close parentheses to the power of 15 space left square bracket 20 space cross times space open parentheses 14 close parentheses cubed space plus space 15 space cross times space left parenthesis 14 right parenthesis squared space plus space 6 space cross times space 14 space plus 1 right square bracket space equals fraction numerator 1 over denominator left parenthesis 15 right parenthesis to the power of 6 end fraction left square bracket 54880 plus 2940 plus 84 plus 1 right square bracket space equals fraction numerator 57905 over denominator left parenthesis 15 right parenthesis to the power of 6 end fraction.$

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Probability

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. Why is the probability that if six randomly selected telephone numbers are called, at least three of them will be busy?

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