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To solve the given problems, let's first outline the formulas used in the situations described:

We are given:
- \( P(A) = 0.51 \)
- \( P(B) = 0.46 \)

### Part (a): When \( A \) and \( B \) are **independent events**:
The formula for the probability of \( A \cup B \) (union of \( A \) and \( B \)) when they are independent is:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

For independent events, the probability of the intersection \( P(A \cap B) \) is:

\[
P(A \cap B) = P(A) \times P(B)
\]

Now we can calculate:

\[
P(A \cup B) = P(A) + P(B) - P(A) \times P(B)
\]

Substituting the given values:

\[
P(A \cup B) = 0.51 + 0.46 - (0.51 \times 0.46)
\]

Let's compute this:

\[
P(A \cup B) = 0.51 + 0.46 - 0.2346 = 0.7354
\]

Thus, \( P(A \cup B) = 0.7354 \) when \( A \) and \( B \) are independent.

### Part (b): When \( A \) and \( B \) are **mutually exclusive events**:
For mutually exclusive events, the probability of the intersection is zero, i.e., \( P(A \cap B) = 0 \). So the formula for the union simplifies to:

\[
P(A \cup B) = P(A) + P(B)
\]

Substituting the values:

\[
P(A \cup B) = 0.51 + 0.46 = 0.97
\]

Thus, \( P(A \cup B) = 0.97 \) when \( A \) and \( B \) are mutually exclusive.

### Final Answers:
- (a) \( P(A \cup B) = 0.7354 \) (independent events)
- (b) \( P(A \cup B) = 0.97 \) (mutually exclusive events)

Would you like a more detailed explanation or have any questions?

Here are 5 related questions:
1. What are other types of relationships between events, besides independence and mutual exclusivity?
2. How would the union probability change if \( P(A \cap B) \) was known but the events weren't independent or exclusive?
3. What happens to \( P(A \cup B) \) when both \( P(A) \) and \( P(B) \) are very small?
4. Can independent events also be mutually exclusive?
5. How does conditional probability affect the calculation of \( P(A \cup B) \)?

**Tip:** Always remember that mutually exclusive events cannot happen simultaneously, so their intersection is always zero.

Hallar P(A ∪ B), dado que P(A) = 0.51 y P(B) = 0.46, y los eventos A y B son (a) independientes y (b) mutuamente excluyentes.

Learn how to calculate P(A ∪ B) when P(A) = 0.51 and P(B) = 0.46, and the events are either independent or mutually exclusive. Understand key probability concepts, including the addition rule and the difference between independent and mutually exclusive events. Suitable for students in Grades 10-12.

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