A person buys a lottery ticket in 50 lotterie | vedclass

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Question

(A) Let $X$ represent the number of winning prizes in $50$ lotteries. The trials are Bernoulli trials. Clearly,$X$ follows a binomial distribution wit

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

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(A) Let $X$ represent the number of winning prizes in $50$ lotteries. The trials are Bernoulli trials. Clearly,$X$ follows a binomial distribution with $n=50$ and $p=\frac{1}{100}$.$	herefore q = 1 - p = 1 - \frac{1}{100} = \frac{99}{100}$.The probability mass function is given by $P(X=x) = ^{n}C_{x} p^{x} q^{n-x} = ^{50}C_{x} \left(\frac{1}{100}ight)^{x} \left(\frac{99}{100}ight)^{50-x}$.We want to find the probability of winning exactly once,which is $P(X=1)$.$P(X=1) = ^{50}C_{1} \left(\frac{1}{100}ight)^{1} \left(\frac{99}{100}ight)^{50-1}$.$P(X=1) = 50 	imes \frac{1}{100} 	imes \left(\frac{99}{100}ight)^{49}$.$P(X=1) = \frac{50}{100} 	imes \left(\frac{99}{100}ight)^{49} = \frac{1}{2} \left(\frac{99}{100}ight)^{49}$.

$A$ 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?

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