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 at least twice.
Solution
Let X denote the number of times the person wins the lottery.
Then, X follows a binomial distribution with n = 50.
\[\text{ Let p be the probability of winning a prize }.\]
\[ \therefore p = \frac{1}{100}, q = 1 - \frac{1}{100} = \frac{99}{100}\]
\[\text{ Hence, the distribution is given by } \]
\[P(X = r) =^{50}{}{C}_r \left( \frac{1}{100} \right)^r \left( \frac{99}{100} \right)^{50 - r} , r = 0, 1, 2 . . . 50\]
\[ P(\text{ winning at lease twice}) = P(X\geq 2)\]
\[ = 1 - P(X = 0) - P(X = 1)\]
\[ = 1 - \left( \frac{99}{100} \right)^{50} - ^{50}{}{C}_1 \times \frac{1}{100} \times \left( \frac{99}{100} \right)^{49} \]
\[ = 1 - \frac{{99}^{49} \times 149}{{100}^{50}}\]
Hence, the probability of winning the prize at least twice \[ = 1 - \frac{{99}^{49} \times 149}{{100}^{50}}\]
