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Probability

Question
CBSEENMA12033917
Wired Faculty App

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1 over 100. What is the probaility that he will win a prize exactly once (a) at least once (b) exactly once (c) at least twice?

Solution

Here n = 50
     straight p space equals 1 over 100 comma space space space straight q space equals space 1 minus space straight p space equals space 1 minus 1 over 100 space equals space 99 over 100
(a) P(at least once) = 1 - P(0) = 1 minus space straight C presuperscript 50 subscript 0 space open parentheses 99 over 100 close parentheses to the power of 50 space equals space 1 minus space open parentheses 99 over 100 close parentheses to the power of 50
(b) P(exactly once) = P(1) = straight C presuperscript 50 subscript 1 space open parentheses 99 over 100 close parentheses to the power of 49 space open parentheses 1 over 100 close parentheses
                             equals space 50 over 1 cross times open parentheses 99 over 100 close parentheses to the power of 49 space cross times space 1 over 100 space equals space 1 half open parentheses 99 over 100 close parentheses to the power of 49.
(c) P(at least twice) = 1 - [P(0) + P(1)]
                         equals space 1 minus space open square brackets straight C presuperscript 50 subscript 0 space open parentheses 99 over 100 close parentheses to the power of 50 space plus space straight C presuperscript 50 subscript 1 space open parentheses 99 over 100 close parentheses to the power of 49 space open parentheses 1 over 100 close parentheses close square brackets space equals space 1 minus space open square brackets open parentheses 99 over 100 close parentheses to the power of 50 space plus space 1 half open parentheses 99 over 100 close parentheses to the power of 49 close square brackets space equals space 1 minus space open parentheses 99 over 100 close parentheses to the power of 50 space minus space 1 half open parentheses 99 over 100 close parentheses to the power of 49 space equals space 1 minus space open parentheses 99 over 100 close parentheses to the power of 49 space open square brackets 99 over 100 plus 1 half close square brackets space equals space 1 minus space open parentheses 99 over 100 close parentheses to the power of 49 space open parentheses fraction numerator 99 plus 50 over denominator 100 end fraction close parentheses space equals space 1 minus 49 over 100 open parentheses 99 over 100 close parentheses to the power of 49.

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