A Person Buys a Lottery Ticket in 50 Lotteries, in Each of Which His Chance of Winning a Prize is 1 100 . What is the Probability that He Will Win a Prize Exactly Once . - Mathematics and Statistics

Sum

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 $=\frac{1}{2} \left( \frac{99}{100} \right)^{49}$

Concept: Bernoulli Trials and Binomial Distribution
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 33 Binomial Distribution
Exercise 33.1 | Q 44.2 | Page 15