Question

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

• A. \( \frac{121}{1024} \)
• B. \( \frac{774}{1024} \)
• C. \( \frac{781}{1024} \)
• D. \( \frac{223}{1024} \)

Hint:

Use the complement rule for problems involving "at least one" events. Find the probability of the complementary event (no wins) and subtract it from 1.

Answer 1

The Correct Option is C

Step 1: Use the complement rule.
The probability of winning at least one prize is the complement of the probability of not winning any prizes. If the probability of winning on a single ticket is \( \frac{1}{4} \), then the probability of not winning on a single ticket is \( 1 - \frac{1}{4} = \frac{3}{4} \).
Step 2: Calculate the probability of not winning on 5 tickets.
The probability of not winning on all 5 tickets is: \[ \left( \frac{3}{4} \right)^5 = \frac{243}{1024}. \]
Step 3: Calculate the probability of winning at least one prize.
The probability of winning at least one prize is the complement: \[ 1 - \frac{243}{1024} = \frac{781}{1024}. \]
Step 4: Conclusion.
Thus, the probability that the person wins at least one prize is \( \frac{781}{1024} \), which corresponds to option (C).