6. What event is complementary to the event that the product is an even number? Determine the probability of both events.

The **complementary event** to the event "the product is an even number" is "the product is an odd number." To determine the probabilities of both events, let's consider the scenario where two numbers are selected independently. For simplicity, we can assume these numbers are chosen from a standard set where each number has an equal probability of being even or odd (e.g., rolling two fair six-sided dice). ### 1. Complementary Event: Product is Odd - **Condition for Odd Product:** Both numbers must be **odd**. - **Probability of Selecting an Odd Number:** If selecting from a set where half the numbers are odd, the probability is \( \frac{1}{2} \) for each number. - **Combined Probability:** \[ P(\text{Product is Odd}) = P(\text{First number is Odd}) \times P(\text{Second number is Odd}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \] ### 2. Original Event: Product is Even - **Condition for Even Product:** At least one of the numbers must be **even**. - **Complementary Calculation:** Since the total probability is 1, and we've calculated the probability of the product being odd: \[ P(\text{Product is Even}) = 1 - P(\text{Product is Odd}) = 1 - \frac{1}{4} = \frac{3}{4} \] ### Summary - **Complementary Event:** The product is an **odd number**. - **Probability of Product Being Even:** \( \frac{3}{4} \) or 75%. - **Probability of Product Being Odd:** \( \frac{1}{4} \) or 25%. This analysis assumes that each number has an equal chance of being odd or even and that the selections are independent. **Answer:** Its complementary event is that the product is odd. The probability the product is even is ¾ and the probability it is odd is ¼.