2. Which of the following numbers on the spinner are greater than 3 ? 4. There is more than one $$ = $$ event of interest. 5. How can you represent the situation as a probability using symbols and variables? 6. What is the sample space? $$ \mathrm{S}=1,1,2,3,4,5,6,7 $$ words and or or.) 7. How many possible outcomes are in the sample space? 8. How many favorable outcomes are in the sample space? How likely is it that Rosette will get a higher number?

Question

UpStudy AI · Accepted Answer

Sure, let's address each of your questions step-by-step based on the spinner with the sample space $$ \mathrm{S} = \{1, 1, 2, 3, 4, 5, 6, 7\} $$.

---

### **2. Which of the following numbers on the spinner are greater than 3?**

**Answer:**  
The numbers on the spinner greater than 3 are:
- **4**
- **5**
- **6**
- **7**

---

### **4. There is more than one event of interest.**

**Explanation:**  
In probability, an *event* is a set of outcomes to which a probability is assigned. Since multiple conditions or outcomes can be of interest, there can be multiple events. For example:
- **Event A:** Spinner lands on a number greater than 3.
- **Event B:** Spinner lands on an odd number.

These are two distinct events of interest derived from the same spinner.

---

### **5. How can you represent the situation as a probability using symbols and variables?**

**Answer:**  
You can represent the probability of an event using the following formula:

$$
P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}
$$

**Using Variables:**
- Let $$ S $$ be the **sample space**.
- Let $$ A $$ be the **event** of interest (e.g., spinning a number greater than 3).

Then,

$$
P(A) = \frac{|A|}{|S|}
$$

Where:
- $$ |A| $$ = number of favorable outcomes for event $$ A $$.
- $$ |S| $$ = total number of possible outcomes in the sample space.

---

### **6. What is the sample space? $$ \mathrm{S} = \{1, 1, 2, 3, 4, 5, 6, 7\} $$**

**Answer:**  
The **sample space** $$ S $$ is the set of all possible outcomes when the spinner is spun. Given:

$$
S = \{1, 1, 2, 3, 4, 5, 6, 7\}
$$

This means there are **8** possible outcomes, with the number **1** appearing twice.

---

### **7. How many possible outcomes are in the sample space?**

**Answer:**  
There are **8** possible outcomes in the sample space $$ S $$.

---

### **8. How many favorable outcomes are in the sample space?**

**Assumption:**  
Assuming the **favorable outcomes** refer to spinning a number **greater than 3**.

**Answer:**  
The favorable numbers greater than 3 are **4, 5, 6,** and **7**. Therefore, there are **4** favorable outcomes.

---

### **Final Question: How likely is it that Rosette will get a higher number?**

**Clarification Needed:**  
To determine how likely it is that Rosette will get a higher number, we need to know **what she is comparing her number to**. However, I'll assume that Rosette is comparing her spin to another independent spin (e.g., another person’s spin) using the same spinner.

**Calculation:**

1. **Sample Space for One Spin:**  
   $$ S = \{1, 1, 2, 3, 4, 5, 6, 7\} $$ with **8** possible outcomes.

2. **Total Possible Pairs of Spins:**  
   Since both Rosette and the other person spin independently, there are $$ 8 \times 8 = 64 $$ possible pairs.

3. **Favorable Outcomes (Rosette's number > Other's number):**  
   - **Rosette spins a 2:**  
     Other can spin **1** (which has 2 occurrences).  
     Favorable pairs: 2
     
   - **Rosette spins a 3:**  
     Other can spin **1** or **2** (total 3 outcomes).  
     Favorable pairs: 3
     
   - **Rosette spins a 4:**  
     Other can spin **1, 2,** or **3** (total 4 outcomes).  
     Favorable pairs: 4
     
   - **Rosette spins a 5:**  
     Other can spin **1, 2, 3,** or **4** (total 5 outcomes).  
     Favorable pairs: 5
     
   - **Rosette spins a 6:**  
     Other can spin **1, 2, 3, 4,** or **5** (total 6 outcomes).  
     Favorable pairs: 6
     
   - **Rosette spins a 7:**  
     Other can spin **1, 2, 3, 4, 5,** or **6** (total 7 outcomes).  
     Favorable pairs: 7

- **Rosette spins a 1:**  
     No favorable outcomes since 1 is the lowest number.  
     Favorable pairs: 0 (but since there are two 1's, total = 0)

4. **Total Favorable Pairs:**  
   $$ 2 + 3 + 4 + 5 + 6 + 7 = 27 $$ favorable pairs.

5. **Probability Calculation:**  
   $$
   P(\text{Rosette's number} > \text{Other's number}) = \frac{\text{Favorable Pairs}}{\text{Total Pairs}} = \frac{27}{64} \approx 0.4219 \text{ or } 42.19\%
   $$

**Answer:**  
There is a **$$\frac{27}{64}$$** (approximately **42.19%**) probability that Rosette will spin a higher number than the other spin.

---

**Summary:**  
- **Sample Space ($$ S $$)**: $$ \{1, 1, 2, 3, 4, 5, 6, 7\} $$
- **Total Outcomes**: 8
- **Favorable Outcomes (Numbers > 3)**: 4
- **Probability of Spinning > 3**: $$ \frac{4}{8} = 0.5 $$ or **50%**
- **Probability Rosette Spins Higher Than Another Spin**: $$ \frac{27}{64} \approx 42.19\% $$

If you have more specific details or need further clarification on any of these points, feel free to ask!
There are 8 possible outcomes in the sample space. The numbers greater than 3 are 4, 5, 6, and 7. The probability that Rosette spins a number greater than 3 is 50%.

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