You and a friend are playing a game in which you take turns rolling a six-sided number cube, and spinning a spinner with 4 congruent sections numbered \( 1-4 \). You and your friend take turns. - If your friend rolls an odd number and spins an even number, he or she wins. - If you roll an even number and spin an odd number, you win. 1. List the possible outcomes for playing the game. 2. Explain why the events of your friend winning or of you winning are compound events. 3. List the outcomes for the event of your friend winning the game and for the event of you winning the game. Then calculate each probability and record in the table. 3era

Let's break down each part of the problem step by step. --- ### **1. List the Possible Outcomes for Playing the Game** In this game, each play involves two actions: - **Rolling a six-sided die** with numbers **1 through 6**. - **Spinning a spinner** divided into **4 congruent sections** numbered **1 through 4**. To list all possible outcomes, we consider every combination of die roll and spinner number. Each outcome can be represented as an ordered pair **(Die Number, Spinner Number)**. Here are all the possible outcomes: | Die Roll | Spinner Spin | Outcome | |----------|--------------|---------| | 1 | 1 | (1, 1) | | 1 | 2 | (1, 2) | | 1 | 3 | (1, 3) | | 1 | 4 | (1, 4) | | 2 | 1 | (2, 1) | | 2 | 2 | (2, 2) | | 2 | 3 | (2, 3) | | 2 | 4 | (2, 4) | | 3 | 1 | (3, 1) | | 3 | 2 | (3, 2) | | 3 | 3 | (3, 3) | | 3 | 4 | (3, 4) | | 4 | 1 | (4, 1) | | 4 | 2 | (4, 2) | | 4 | 3 | (4, 3) | | 4 | 4 | (4, 4) | | 5 | 1 | (5, 1) | | 5 | 2 | (5, 2) | | 5 | 3 | (5, 3) | | 5 | 4 | (5, 4) | | 6 | 1 | (6, 1) | | 6 | 2 | (6, 2) | | 6 | 3 | (6, 3) | | 6 | 4 | (6, 4) | **Total Possible Outcomes:** 6 (die) × 4 (spinner) = **24 outcomes** --- ### **2. Why Are the Events of Winning Compound Events?** A **compound event** is an event that consists of **two or more simple events** happening together. - **For Your Friend to Win:** - **Rolling an Odd Number on the Die** *(Simple Event 1)*: Possible numbers are 1, 3, or 5. - **Spinning an Even Number on the Spinner** *(Simple Event 2)*: Possible numbers are 2 or 4. Both Simple Event 1 **and** Simple Event 2 must occur simultaneously for your friend to win. - **For You to Win:** - **Rolling an Even Number on the Die** *(Simple Event 1)*: Possible numbers are 2, 4, or 6. - **Spinning an Odd Number on the Spinner** *(Simple Event 2)*: Possible numbers are 1 or 3. Both Simple Event 1 **and** Simple Event 2 must occur simultaneously for you to win. Since winning requires **combining two independent events**, both your friend's winning condition and your winning condition are **compound events**. --- ### **3. Outcomes for Winning Events and Their Probabilities** Let's identify the specific outcomes that lead to either you or your friend winning and then calculate their probabilities. #### **a. Outcomes Where Your Friend Wins** **Winning Conditions:** - **Die Roll:** Odd numbers (1, 3, 5) - **Spinner Spin:** Even numbers (2, 4) **Favorable Outcomes:** | Die Roll | Spinner Spin | Outcome | |----------|--------------|---------| | 1 | 2 | (1, 2) | | 1 | 4 | (1, 4) | | 3 | 2 | (3, 2) | | 3 | 4 | (3, 4) | | 5 | 2 | (5, 2) | | 5 | 4 | (5, 4) | **Total Favorable Outcomes for Friend:** 6 **Probability Calculation:** \[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{6}{24} = \frac{1}{4} = 25\% \] #### **b. Outcomes Where You Win** **Winning Conditions:** - **Die Roll:** Even numbers (2, 4, 6) - **Spinner Spin:** Odd numbers (1, 3) **Favorable Outcomes:** | Die Roll | Spinner Spin | Outcome | |----------|--------------|---------| | 2 | 1 | (2, 1) | | 2 | 3 | (2, 3) | | 4 | 1 | (4, 1) | | 4 | 3 | (4, 3) | | 6 | 1 | (6, 1) | | 6 | 3 | (6, 3) | **Total Favorable Outcomes for You:** 6 **Probability Calculation:** \[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{6}{24} = \frac{1}{4} = 25\% \] #### **c. Summary Table** | Event | Number of Favorable Outcomes | Probability | |-------------------------|------------------------------|-------------| | **Friend Wins** | 6 | 25% | | **You Win** | 6 | 25% | --- **Note:** The remaining 12 outcomes (6 + 6) represent scenarios where neither you nor your friend wins. These could occur if: - Your friend rolls an odd number and spins an odd number. - Your friend rolls an even number and spins an even number. - You roll an odd number and spin an even number. - You roll an even number and spin an even number. However, these are beyond the scope of the current questions. ---