4. Which events from Question 3 represent complementary events? Explain your reasoning. 5. Betina says that the product being less than 10 and the product being more than 10 are complementary events. Davika disagrees. Who is correct? Explain your reasoning

To address your questions effectively, let's first establish some context based on the information provided: - **Question 3** likely involves calculating the probability of events related to the product of two numbers (e.g., rolling two dice and multiplying the outcomes). - **Question 4** asks which events from Question 3 are complementary. - **Question 5** involves a discussion between Betina and Davika about whether the events "the product being less than 10" and "the product being more than 10" are complementary. Let's tackle each question step-by-step. --- ### **4. Which events from Question 3 represent complementary events? Explain your reasoning.** **Answer:** **Complementary events** are two outcomes that are mutually exclusive (they cannot happen at the same time) and together cover all possible outcomes of an experiment. In other words, the probability of one event occurring plus the probability of the other event occurring equals 1. Assuming **Question 3** involves calculating the product of two numbers (for example, rolling two six-sided dice and multiplying the results), complementary events would need to cover all possible outcomes without overlapping. **Possible Complementary Events:** 1. **Product is less than or equal to 10 (`P ≤ 10`).** 2. **Product is greater than 10 (`P > 10`).** These two events are: - **Mutually Exclusive:** A product cannot be both ≤10 and >10 simultaneously. - **Collectively Exhaustive:** Together, they cover all possible products (since every product is either ≤10 or >10). **Reasoning:** - If the product is **≤10**, it cannot be **>10**, and vice versa. - There are no other possibilities outside these two categories for the product. Therefore, **"Product is less than or equal to 10"** and **"Product is greater than 10"** are complementary events. --- ### **5. Betina says that the product being less than 10 and the product being more than 10 are complementary events. Davika disagrees. Who is correct? Explain your reasoning.** **Answer:** **Davika is correct.** Betina's assertion that **"the product being less than 10"** and **"the product being more than 10"** are complementary events is **incorrect**. Here's why: **Why They Are Not Complementary:** 1. **Exclusion of the Product Being Exactly 10:** - The events **"product < 10"** and **"product > 10"** do **not** account for the scenario where **"product = 10."** - This means there's an outcome where neither event occurs, violating the requirement that complementary events cover all possible outcomes. 2. **Probability Sum Less Than 1:** - Since the scenario where the product equals 10 is omitted, the sum of the probabilities of the two events **"product < 10"** and **"product > 10"** is **less than 1**. - Complementary events must have probabilities that add up to exactly 1. **Correct Complementary Events:** To form complementary events, you should include all possible outcomes without overlap. For example: - **"Product is less than or equal to 10" (`P ≤ 10`).** - **"Product is greater than 10" (`P > 10`).** These two events are complementary because: - They are mutually exclusive. - Together, they cover all possible outcomes (including when the product is exactly 10). **Conclusion:** - **Betina's Claim:** Incorrect, because it omits the outcome where the product is exactly 10. - **Davika's Stance:** Correct, as she recognizes that Betina's events do not account for all possible outcomes, and thus they are not truly complementary. --- **Summary:** - **Question 4:** The complementary events are **"product is less than or equal to 10"** and **"product is greater than 10."** - **Question 5:** **Davika is correct**; the events **"product < 10"** and **"product > 10"** are **not** complementary because they exclude the possibility of the product being exactly 10.