20 Draw a spinner for which $$ \operatorname{Pr}( $$ green $$ )=2 \operatorname{Pr}( $$ blue $$ ) $$ $$ \operatorname{Pr}( $$ blue $$ )=2 \operatorname{Pr}( $$ brown $$ ) $$ $$ \operatorname{Pr}( $$ brown $$ )=2 \operatorname{Pr}( $$ pink $$ ) $$ and $$ \operatorname{Pr}( $$ pink $$ )=2 \operatorname{Pr}( $$ red $$ ) $$

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To design a spinner that satisfies the given probability relationships, follow these steps:

### **Given Relationships:**
1. $$ \Pr(\text{Green}) = 2 \Pr(\text{Blue}) $$
2. $$ \Pr(\text{Blue}) = 2 \Pr(\text{Brown}) $$
3. $$ \Pr(\text{Brown}) = 2 \Pr(\text{Pink}) $$
4. $$ \Pr(\text{Pink}) = 2 \Pr(\text{Red}) $$

### **Step 1: Assign Variables**
Let the probability of landing on **Red** be $$ x $$.

Then, based on the relationships:
- $$ \Pr(\text{Pink}) = 2x $$
- $$ \Pr(\text{Brown}) = 2 \times \Pr(\text{Pink}) = 4x $$
- $$ \Pr(\text{Blue}) = 2 \times \Pr(\text{Brown}) = 8x $$
- $$ \Pr(\text{Green}) = 2 \times \Pr(\text{Blue}) = 16x $$

### **Step 2: Determine the Value of $$ x $$**
Since the total probability must sum to 1:
$$
x + 2x + 4x + 8x + 16x = 31x = 1 \
\Rightarrow x = \frac{1}{31}
$$

### **Step 3: Calculate Each Probability**
- $$ \Pr(\text{Red}) = x = \frac{1}{31} $$
- $$ \Pr(\text{Pink}) = 2x = \frac{2}{31} $$
- $$ \Pr(\text{Brown}) = 4x = \frac{4}{31} $$
- $$ \Pr(\text{Blue}) = 8x = \frac{8}{31} $$
- $$ \Pr(\text{Green}) = 16x = \frac{16}{31} $$

### **Step 4: Translate Probabilities to Angles**
A spinner is a circle of 360 degrees. The angle for each color sector is calculated by multiplying the probability by 360°.

- **Red:** $$ \frac{1}{31} \times 360 \approx 11.61^\circ $$
- **Pink:** $$ \frac{2}{31} \times 360 \approx 23.23^\circ $$
- **Brown:** $$ \frac{4}{31} \times 360 \approx 46.45^\circ $$
- **Blue:** $$ \frac{8}{31} \times 360 \approx 92.90^\circ $$
- **Green:** $$ \frac{16}{31} \times 360 \approx 185.81^\circ $$

### **Step 5: Draw the Spinner**

1. **Draw a Circle:** Start with a circle representing the spinner.
2. **Divide the Circle:** Using a protractor, divide the circle into sectors based on the calculated angles:
   - **Red:** ~11.6°
   - **Pink:** ~23.2°
   - **Brown:** ~46.5°
   - **Blue:** ~92.9°
   - **Green:** ~185.8°
3. **Color Each Sector:** Assign the respective colors to each sector:
   - The largest sector (~185.8°) will be **Green**.
   - Followed by **Blue**, **Brown**, **Pink**, and the smallest sector will be **Red**.

### **Visual Representation:**

```
[Imagine a circle divided into five sectors with the following approximate angles:]
- Green: 185.8°
- Blue: 92.9°
- Brown: 46.5°
- Pink: 23.2°
- Red: 11.6°
```

Each sector's size corresponds to its probability, ensuring the spinner behaves as specified.
To create the spinner, divide the circle into five sectors with the following angles: - **Green:** 185.8° - **Blue:** 92.9° - **Brown:** 46.5° - **Pink:** 23.2° - **Red:** 11.6°

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