\( S=1,2,3,4,5,6,7 \) 5. How many possible outcomes are in the sample space? 6. How many favorable outcomes are in the sample space? ow likely is it that Tucker will get a higher number than Rosette?

Let's analyze the problem step by step based on the sample space \( S = \{1, 2, 3, 4, 5, 6, 7\} \). ### 5. How many possible outcomes are in the sample space? When considering the outcomes for both Tucker and Rosette, each has 7 possible choices (numbers 1 through 7). Since their selections are independent of each other, the total number of possible outcomes in the sample space is calculated by multiplying the number of choices for each: \[ \text{Total possible outcomes} = 7 \times 7 = 49 \] **Answer:** There are **49** possible outcomes in the sample space. --- ### 6. How many favorable outcomes are in the sample space? A **favorable outcome** in this context is when Tucker selects a higher number than Rosette. To find the number of such favorable outcomes, we can systematically count the possibilities: 1. **Rosette selects 1:** Tucker can select any number from 2 to 7. - **Favorable outcomes:** 6 (numbers 2, 3, 4, 5, 6, 7) 2. **Rosette selects 2:** Tucker can select any number from 3 to 7. - **Favorable outcomes:** 5 (numbers 3, 4, 5, 6, 7) 3. **Rosette selects 3:** Tucker can select any number from 4 to 7. - **Favorable outcomes:** 4 (numbers 4, 5, 6, 7) 4. **Rosette selects 4:** Tucker can select any number from 5 to 7. - **Favorable outcomes:** 3 (numbers 5, 6, 7) 5. **Rosette selects 5:** Tucker can select any number from 6 to 7. - **Favorable outcomes:** 2 (numbers 6, 7) 6. **Rosette selects 6:** Tucker can select only the number 7. - **Favorable outcomes:** 1 (number 7) 7. **Rosette selects 7:** There are no higher numbers left for Tucker. - **Favorable outcomes:** 0 Adding these up: \[ 6 + 5 + 4 + 3 + 2 + 1 + 0 = 21 \] **Answer:** There are **21** favorable outcomes in the sample space. --- ### How likely is it that Tucker will get a higher number than Rosette? The probability \( P \) of Tucker selecting a higher number than Rosette is the ratio of favorable outcomes to the total possible outcomes. \[ P(\text{Tucker > Rosette}) = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}} = \frac{21}{49} \] Simplifying the fraction: \[ \frac{21}{49} = \frac{3}{7} \approx 0.4286 \text{ or } 42.86\% \] **Answer:** There is a **\(\frac{3}{7}\) chance (approximately 42.86%)** that Tucker will get a higher number than Rosette. --- ### Summary 1. **Total possible outcomes:** 49 2. **Favorable outcomes (Tucker > Rosette):** 21 3. **Probability Tucker > Rosette:** \(\frac{3}{7}\) (≈42.86%)