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The problem involves calculating the expected value of one spin for a spinner with two regions: Region A and Region B. The central angle for Region A is given as \(7^\circ\), and the values assigned to each region are as follows:

- Region A has a value of 100.
- Region B has a value of 2.

The goal is to calculate the expected value of one spin using two different methods.

### Step 1: Calculate the Probabilities for Each Region
The total angle of the spinner is \(360^\circ\). The probability of landing in each region is proportional to the central angle of that region.

- **Region A** has an angle of \(7^\circ\).
  \[
  P(\text{Region A}) = \frac{7^\circ}{360^\circ} = \frac{7}{360}
  \]

- **Region B** has an angle of \(360^\circ - 7^\circ = 353^\circ\).
  \[
  P(\text{Region B}) = \frac{353^\circ}{360^\circ} = \frac{353}{360}
  \]

### Step 2: Method 1 - Using the Formula for Expected Value
The expected value \(E\) for one spin is the sum of the products of each region's value and its probability:
\[
E = \left( \text{Value of Region A} \times P(\text{Region A}) \right) + \left( \text{Value of Region B} \times P(\text{Region B}) \right)
\]

Substitute the known values:
\[
E = \left( 100 \times \frac{7}{360} \right) + \left( 2 \times \frac{353}{360} \right)
\]

### Step 3: Method 2 - Approximation Based on Geometry
In the second method, you can approximate the expected value by considering the relative size of the two regions and the large difference between their values (100 vs. 2). Since Region B occupies a much larger portion of the spinner, the expected value will be closer to 2. A simple estimate can be made by averaging the extreme values, weighted by the approximate probabilities.

Now let's calculate these values for both methods.

I'll calculate that for you.### Method 1: Calculation of Expected Value
Using the formula for expected value, the result is:
\[
E = 3.91 \quad (\text{rounded to two decimal places})
\]

This is the expected value based on the actual probabilities of landing in each region.

### Method 2: Approximation Method
In this case, we can estimate the expected value to be closer to 2, given that Region B (with the value of 2) occupies most of the spinner (approximately 98% of the area), while Region A (with the value of 100) occupies only about 2%. Therefore, the expected value is dominated by the influence of Region B, which further supports the calculated result being close to 2 but slightly higher due to the presence of Region A's larger value.

Would you like further details or clarifications? Here are 5 related questions to deepen your understanding:

1. How would the expected value change if the angle of Region A were increased?
2. What happens to the expected value if Region A's value changes to a lower number?
3. Can you derive the variance and standard deviation for this spinner's outcome?
4. How does this problem relate to real-world applications of expected value, such as gambling or insurance?
5. What would the expected value be if the spinner were divided into more regions?

**Tip**: When dealing with probability problems involving geometrical shapes, always consider how the proportions of the shape (e.g., angles) influence the likelihood of outcomes.

Examine the spinner at right. If the central angle of Region A is 7°, calculate the expected value of one spin using two different methods.

Learn how to calculate the expected value of one spin for a spinner with two regions. Region A has a central angle of 7° and a value of 100, while Region B has a value of 2. This problem covers key probability concepts and expected value calculation.

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