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### Question 10:
We are given the following probabilities:
- Probability of a student taking a math class: \( P(\text{Math}) = 0.76 \)
- Probability of a student taking an English class: \( P(\text{English}) = 0.80 \)
- Probability of a student taking both math and English: \( P(\text{Math and English}) = 0.76 \)

We are tasked with finding the probability that a randomly selected student is taking either a math class or an English class, i.e., \( P(\text{Math or English}) \).

Using the formula for the union of two events:
\[
P(\text{Math or English}) = P(\text{Math}) + P(\text{English}) - P(\text{Math and English})
\]
Substitute the values:
\[
P(\text{Math or English}) = 0.76 + 0.80 - 0.76 = 0.80
\]

Thus, the probability that a randomly selected student is taking a math class or an English class is **0.80**.

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### Question 11:
We are given the following probabilities:
- \( P(A) = 0.35 \)
- \( P(A \text{ or } B) = 0.80 \)
- \( P(A \text{ and } B) = 0.20 \)

We need to find \( P(B) \).

Using the formula for the union of two events:
\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\]
Rearranging the formula to solve for \( P(B) \):
\[
P(B) = P(A \text{ or } B) - P(A) + P(A \text{ and } B)
\]
Substitute the given values:
\[
P(B) = 0.80 - 0.35 + 0.20 = 0.65
\]

Thus, the probability \( P(B) \) is **0.65**.

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Would you like further details on these calculations, or do you have any other questions? Here are five related questions you might find useful:

1. How do we calculate the probability of the union of two events in general?
2. What is the difference between independent and mutually exclusive events?
3. How would the answer change if the events were independent?
4. Can we use Venn diagrams to visually solve problems like this?
5. What happens if \( P(A \text{ and } B) = 0 \)? How would the formula simplify?

**Tip:** Always ensure to check if events are mutually exclusive before using formulas, as this can simplify the calculation of their union.

In a large school, it was found that 76% of students are taking a math class, 80% of students are taking an English class, and 76% of students are taking both. Find the probability that a randomly selected student is taking a math class or an English class. Also, find the probability of event B given P(A) = 0.35, P(A or B) = 0.80, and P(A and B) = 0.20.

Learn how to calculate the probability of a randomly selected student taking either a math class or an English class in a large school, as well as determining P(B) using the given probabilities. This problem covers key probability concepts and formulas. Suitable for high school students.

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