Question: A school organizes a lucky draw where students can win either a gift card ( \( 40 \% \) chance) or a book ( \( 25 \% \) chance). There is also a \( 15 \% \) chance that a student wins both. (a) What is the probability that a randomly selected student wins at least one prize? Are winning a gift card and winning a book mutually exclusive? (b) Suppose two students participate in the lucky draw. If prizes are awarded independently, what is the probability that both students win a gift card? How would this probability change if, after the first student wins a prize, the prize is removed and not replaced before the second student's draw?

To find the probability that a randomly selected student wins at least one prize, we can use the inclusion-exclusion principle. The probability of winning at least one prize is given by: \[ P(G \cup B) = P(G) + P(B) - P(G \cap B) \] Where: - \( P(G) = 0.40 \) (probability of winning a gift card) - \( P(B) = 0.25 \) (probability of winning a book) - \( P(G \cap B) = 0.15 \) (probability of winning both) Substituting the values: \[ P(G \cup B) = 0.40 + 0.25 - 0.15 = 0.50 \] So, there is a 50% chance that a student wins at least one prize. Winning a gift card and winning a book are not mutually exclusive since there is a chance that a student can win both. For part (b), the probability that both students win a gift card (assuming independence) is: \[ P(G) \times P(G) = 0.40 \times 0.40 = 0.16 \] If the first student wins a gift card and the prize is removed before the second student's draw, the probability for the second student becomes: - There's one less gift card available, but since prizes are considered independently and we assume the total number remains the same (as the total number of students is not specified), the second student's probability remains at 0.40 since he is part of the same independent draw. Thus, the probability remains unchanged at 0.16, unless stated otherwise about the total number of gift cards and students.