The probability that a person wins a prize on | vedclass

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(C) Let $n = 5$ be the number of lottery tickets purchased. Let $p$ be the probability of winning a prize on a single ticket,so $p = \frac{1}{4}$. Let

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$\frac{781}{1024}$

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(C) Let $n = 5$ be the number of lottery tickets purchased. Let $p$ be the probability of winning a prize on a single ticket,so $p = \frac{1}{4}$. Let $q$ be the probability of not winning a prize on a single ticket,so $q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4}$. We want to find the probability of winning at least one prize,which is $P(X \ge 1)$. Using the complement rule,$P(X \ge 1) = 1 - P(X = 0)$. The probability of winning zero prizes in $5$ trials is given by the binomial distribution formula $P(X = k) = \binom{n}{k} p^k q^{n-k}$. For $k = 0$,$P(X = 0) = \binom{5}{0} (\frac{1}{4})^0 (\frac{3}{4})^5 = 1 	imes 1 	imes \frac{243}{1024} = \frac{243}{1024}$. Therefore,$P(X \ge 1) = 1 - \frac{243}{1024} = \frac{1024 - 243}{1024} = \frac{781}{1024}$.

The probability that a person wins a prize on a lottery ticket is $\frac{1}{4}$. If he purchases $5$ lottery tickets at random,then the probability that he wins at least one prize is

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