Suppose A and B are events such that P(A) = 0.78 and P(B) = 0.17 .
Assuming that events A and B are mutually exclusive, help Luca answer the following questions. You may round your answers to 2 decimal places, and use previous values in subsequent computations.
Showing work is REQUIRED.
1. P(ANB) =
2. P(AUB) =
3. P(A |B) =

Let's help Luca with the following probability questions based on the given information that events \( A \) and \( B \) are mutually exclusive.

### Given:
- \( P(A) = 0.78 \)
- \( P(B) = 0.17 \)
- Since \( A \) and \( B \) are **mutually exclusive**, they cannot both occur simultaneously, meaning \( P(A \cap B) = 0 \).

### 1. \( P(A \cap B) \) (Intersection of A and B):

Since \( A \) and \( B \) are mutually exclusive, the probability that both \( A \) and \( B \) occur at the same time is zero.

\[
P(A \cap B) = 0
\]

### 2. \( P(A \cup B) \) (Union of A and B):

The probability of the union of two mutually exclusive events is the sum of their individual probabilities. Since \( P(A \cap B) = 0 \), the formula simplifies to:

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

Substituting the values:

\[
P(A \cup B) = 0.78 + 0.17 = 0.95
\]

### 3. \( P(A | B) \) (Conditional Probability of A given B):

The conditional probability formula is:

\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]

Since \( P(A \cap B) = 0 \) (because the events are mutually exclusive), we have:

\[
P(A | B) = \frac{0}{0.17} = 0
\]

### Summary:
1. \( P(A \cap B) = 0 \)
2. \( P(A \cup B) = 0.95 \)
3. \( P(A | B) = 0 \)

Would you like further clarification on any of these steps?

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Here are 5 related questions to consider:

1. What is \( P(B | A) \), given that \( A \) and \( B \) are mutually exclusive?
2. How would \( P(A \cup B) \) change if \( A \) and \( B \) were **not** mutually exclusive?
3. What would \( P(A \cap B) \) be if \( A \) and \( B \) were independent?
4. How does the fact that \( A \) and \( B \) are mutually exclusive affect the calculation of \( P(A \cup B) \)?
5. Can two independent events ever be mutually exclusive?

**Tip**: For mutually exclusive events, the intersection probability is always zero, simplifying many calculations!

Suppose A and B are events such that P(A) = 0.78 and P(B) = 0.17. Assuming that events A and B are mutually exclusive, help Luca answer the following questions. You may round your answers to 2 decimal places, and use previous values in subsequent computations. Showing work is REQUIRED.
1. P(ANB) =
2. P(AUB) =
3. P(A |B) =

Solve a probability problem involving mutually exclusive events where P(A) = 0.78 and P(B) = 0.17. Learn to compute P(A ∩ B), P(A ∪ B), and P(A | B) step-by-step, using key probability concepts. Suitable for students in Grades 10-12.

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