The probability that a person wins a prize on a lottery ticket is $\frac{1}{4} .$ If he purchases 5 lottery…

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
  1. $\frac{121}{1024}$
  2. $\frac{774}{1024}$
  3. $\frac{781}{1024}$
  4. $\frac{223}{1024}$

Solution

Given that the probability of a person wins a prize on a lottery ticket is \(P=\frac{1}{4}\) and the probability of a person loose a prize on a lottery ticket is \(q=\frac{3}{4}\).
Therefore, the probability that he wins at least one prize is
\(\begin{aligned}
& =\left(\frac{1}{1024}\right)+\left(\frac{1}{256}\right)\left(\frac{3}{4}\right)+\left(\frac{1}{64}\right)\left(\frac{9}{16}\right)+\left(\frac{1}{16}\right)\left(\frac{27}{64}\right)+\left(\frac{1}{4}\right)\left(\frac{81}{256}\right) \\
& =\frac{781}{1024}
\end{aligned}\)

Asked in: MHT CET 2020 (12 Oct Shift 1)

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