Question

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning prize is $\frac{1}{100}$. What is the probability that he will win a prize.
exactly once ?

Answer 1

Let X denotes the number of times that the person wins the prize, Here, it is a case of Bernoulli trails with n=50, p=1/100 and q=1-p=1-$\frac{1}{100}$=$\frac{99}{100}$.
$\therefore$ P(X=r)= ${}^n{C}_r{p}^r{q}^{n-r}={}^{50}{C}_r{\left(\frac{1}{100}\right)}^r.{\left(\frac{99}{100}\right)}^{50-r}$

P (he wins a prize exactly once)

=${}^{50}{C}_1{p}^1{q}^{49}=50\left(\frac{1}{100}\right){\left(\frac{99}{100}\right)}^{49}=\frac{1}{2}{\left(\frac{99}{100}\right)}^{49}$