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Question
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \[\frac{1}{100} .\] What is the probability that he will win a prize exactly once.
Solution
Let X denote the number of times the person wins the lottery.
Then, X follows a binomial distribution with n = 50.
\[\text{ Let p be the probability of winning a prize } . \]
\[ \therefore p = \frac{1}{100}, q = 1 - \frac{1}{100} = \frac{99}{100}\]
\[\text{ Hence, the distribution is given by } \]
\[P(X = r) =^{50}{}{C}_r \left( \frac{1}{100} \right)^r \left( \frac{99}{100} \right)^{50 - r} , r = 0, 1, 2 . . . 50\]
\[P(\text{winning exactly once}) = P(X = 1)\]
\[ = ^{50}{}{C}_1 \left( \frac{1}{100} \right)^1 \left( \frac{99}{100} \right)^{50 - 1} \]
\[ = \frac{1}{2} \left( \frac{99}{100} \right)^{49} \]
Hence, probability of winning a prize exactly once \[=\frac{1}{2} \left( \frac{99}{100} \right)^{49}\]
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