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Probability

Question
CBSEENMA12033885

Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in a single die is considered a success, find the probability of at least three successes.

Solution

n = 5
p = P(3 or 4 or 5) = 3 over 6 space equals 1 half comma space space straight q space equals space 1 minus straight p space equals space 1 minus 1 half space equals space 1 half
P(at least three successes) = P(3) + P(4) + P(5)
                             equals space straight C presuperscript 5 subscript 3 space open parentheses 1 half close parentheses squared space open parentheses 1 half close parentheses cubed space plus space straight C presuperscript 5 subscript 4 space open parentheses 1 half close parentheses space open parentheses 1 half close parentheses to the power of 4 plus space straight C presuperscript 5 subscript 5 space open parentheses 1 half close parentheses to the power of 5
equals space open parentheses 1 half close parentheses to the power of 5 space open square brackets straight C presuperscript 5 subscript 3 space plus space straight C presuperscript 5 subscript 4 space plus space straight C presuperscript 5 subscript 5 close square brackets space equals space 1 over 32 open square brackets fraction numerator 5 space cross times 4 over denominator 1 cross times 2 end fraction plus 5 over 1 plus 1 close square brackets
equals space 1 over 32 left parenthesis 10 plus 5 plus 1 right parenthesis space equals space 16 over 32 equals space 1 half

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