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? Note: The answer should be in detail explaining which method is best. Students must respond to at least two classmates regarding their thoughts on the answers.

To find the probability that a randomly selected student wins at least one prize, we can use the principle of inclusion-exclusion. Let's denote: - \( P(G) = 0.40 \) (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). The probability that a student wins at least one prize is: \[ P(G \cup B) = P(G) + P(B) - P(G \cap B) = 0.40 + 0.25 - 0.15 = 0.50. \] So, there’s a 50% chance a student wins at least one prize! Winning a gift card and winning a book are not mutually exclusive since there is a chance of winning both simultaneously — hence, they are dependent events. Now, regarding two students participating independently, the probability that both win a gift card is simply the product of the individual probabilities: \[ P(\text{Both win G}) = P(G) \times P(G) = 0.40 \times 0.40 = 0.16. \] If the first student wins a gift card and it's removed before the second draw, we assume now that the probabilities may adjust. If only one gift card is available, the second student has only a chance of 0.25 (winning a book) and 0 probability of winning a gift card since it's removed. Thus, the probability both could win a gift card in this modified scenario would be 0. To summarize, it’s crucial to first calculate whether events are overlapping or mutually exclusive, which leads to using the right probability rules. Independence or dependency affects calculations significantly. Remember, no prize replacement changes the game entirely! Getting these probabilities right can help students get ahead in understanding practical applications of probability, whether it's in competitions like this lucky draw or real-world scenarios like assessing risks!