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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?
(a) at least once, (b) exactly once, (c) at least twice?
(a) at least once, (b) exactly once, (c) at least twice?
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Probability that the person wins the prize $=\frac{1}{100}$ Probability of losing $=1-\frac{1}{100}=\frac{99}{100}$
(a) Probability that he loses in all the loteries $=\left(\frac{99}{100}\right)^{50}$ Probability that he wins at least in one lottery $=1-\left(\frac{99}{100}\right)^{50}=1-(.99)^{50}$
(b) Probability that he wins exactly once
$$
={ }^{50} \mathrm{C}_1\left(\frac{99}{100}\right)^{49}\left(\frac{1}{100}\right)^1=\frac{1}{2}\left(\frac{99}{100}\right)^{49}
$$
(c) Probability that he wins at least twice
$$
\begin{aligned}
&=\mathrm{P}(2)+\mathrm{P}(3)+\ldots \ldots+\mathrm{P}(50) \\
&\quad=[\mathrm{P}(0)+\mathrm{P}(1)+\ldots \ldots+\mathrm{P}(50)]-[\mathrm{P}(0)+\mathrm{P}(1)] \\
&=1-[\mathrm{P}(0)+\mathrm{P}(1)]
\end{aligned}
$$
(a) Probability that he loses in all the loteries $=\left(\frac{99}{100}\right)^{50}$ Probability that he wins at least in one lottery $=1-\left(\frac{99}{100}\right)^{50}=1-(.99)^{50}$
(b) Probability that he wins exactly once
$$
={ }^{50} \mathrm{C}_1\left(\frac{99}{100}\right)^{49}\left(\frac{1}{100}\right)^1=\frac{1}{2}\left(\frac{99}{100}\right)^{49}
$$
(c) Probability that he wins at least twice
$$
\begin{aligned}
&=\mathrm{P}(2)+\mathrm{P}(3)+\ldots \ldots+\mathrm{P}(50) \\
&\quad=[\mathrm{P}(0)+\mathrm{P}(1)+\ldots \ldots+\mathrm{P}(50)]-[\mathrm{P}(0)+\mathrm{P}(1)] \\
&=1-[\mathrm{P}(0)+\mathrm{P}(1)]
\end{aligned}
$$
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