6. What event is cemplementary to the event that the product is an even number? Determine the probability of beth events, 7. What is the sum of the probabilities of two complementary events? Explain why your answer makes sense.

Certainly! Let's tackle each question step by step. --- ### **6. Complementary Event to "Product is an Even Number" and Determining Their Probabilities** **a. Identifying the Complementary Event:** - **Original Event:** The product of two numbers is an even number. - **Complementary Event:** The product of two numbers is an **odd** number. **Why?** - In probability, complementary events encompass all possible outcomes that are not part of the original event. Since a product of two integers is either even or odd, these are complementary to each other. **b. Determining the Probabilities:** Let's consider a common scenario where this question might arise: **Rolling two fair six-sided dice**. 1. **Total Possible Outcomes:** - When rolling two dice, each die has 6 faces. - Total outcomes = 6 (first die) × 6 (second die) = **36** possible outcomes. 2. **Calculating Probability that the Product is Odd:** - For the product to be odd, **both numbers rolled must be odd**. - Odd numbers on a die: 1, 3, 5 (3 possibilities per die). - Number of outcomes where both dice show odd numbers = 3 (first die) × 3 (second die) = **9**. - Probability (Product is Odd) = Number of favorable outcomes / Total outcomes = 9/36 = **1/4** or **25%**. 3. **Calculating Probability that the Product is Even:** - Since the complementary event encompasses all other outcomes, we can use: - **Probability (Product is Even) = 1 - Probability (Product is Odd)** - Probability (Product is Even) = 1 - 1/4 = **3/4** or **75%**. --- ### **7. Sum of the Probabilities of Two Complementary Events** **Statement:** - The sum of the probabilities of two complementary events is always **1**. **Explanation:** - **Complementary Events Defined:** Two events are complementary if they are the only two possible outcomes, and they cannot occur simultaneously. In other words, one must occur, and the other cannot. - **Mathematically:** - Let Event A be "Product is Even." - Let Event B be "Product is Odd." - These are complementary: \( P(A) + P(B) = 1 \). - **Reasoning:** - Since the product must be either even or odd (no other possibilities exist in this context), the probabilities of these two exclusive events cover the entire sample space. - Therefore, adding their probabilities accounts for all possible outcomes, totaling to certainty, which is represented by the probability **1**. **Conclusion:** - This principle ensures that all possible outcomes are accounted for in probability calculations, maintaining the foundational axiom that the sum of all probabilities in a sample space is 1. --- I hope this clarifies both the concept of complementary events and how their probabilities relate!