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To solve this problem, we need to calculate the expected value of the **midterm grade** given that the student made exactly **one excuse**. Here's the step-by-step process:

### Step 1: Understanding the Given Data
- The relevant data in the table concerns students who made exactly **one excuse**.
- For **one excuse**, we have:
  - **Midterm Score**: 87
  - **Grades (for Final)**: 86
  - **Probability**: 41% (for Midterm)

### Step 2: Use the Definition of Expected Value
The expected value \( E(X) \) for a given scenario is calculated as:

\[
E(X) = \sum \text{(Value of Outcome)} \times \text{(Probability of Outcome)}
\]

In this case, since we are given only one outcome for the midterm grade with one excuse, the expected value is just the midterm score for that specific condition multiplied by the probability of occurrence.

### Step 3: Compute the Expected Midterm Grade
Since the student made exactly one excuse, the expected midterm grade is simply:

\[
E(\text{Midterm Grade}) = 87 \quad (\text{as there is only one possible grade with one excuse}).
\]

Thus, the expected midterm grade is **87**.

Let me know if you need more details or clarifications. Here are 5 related questions:

1. How would the expected grade change if more outcomes (excuses) were considered?
2. How do we calculate the overall expected value with multiple probabilities?
3. What is the relationship between probabilities and outcomes in statistical expectation?
4. Can the same method be applied to determine the expected final grade?
5. What are real-life scenarios where expected values are useful?

**Tip:** Always verify that the probabilities you use sum up to 1 or are appropriately scaled for the expected value calculation.

Determine the expected value of the midterm grade of a student given that the student made exactly one excuse. Data provided includes the number of excuses, midterm grade probabilities, and final grade probabilities.

This problem involves calculating the expected value of midterm grades, given a student made exactly one excuse. It applies key statistical concepts such as expected value and conditional probability. Suitable for high school students in Grades 10-12.

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