$$ S=1,2,3,4,5,6,7 $$ 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?

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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%)
There are 49 possible outcomes in the sample space. There are 21 favorable outcomes where Tucker selects a higher number than Rosette. The probability that Tucker gets a higher number than Rosette is 3/7, or approximately 42.86%.

\( S=1,2,3,4,5,6,7 \) 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?

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