https://file.aiquickdraw.com/mathbot/file/1727066180049d61wjev3.jpg

Let's solve the given problems step-by-step.

### (a) Compute \( P(A \text{ and } B) \).

We use the formula for the union of two events:

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

Plugging in the values provided:

\[
0.8 = 0.6 + 0.4 - P(A \cap B)
\]

Solving for \( P(A \cap B) \):

\[
P(A \cap B) = 0.6 + 0.4 - 0.8 = 0.2
\]

So, \( P(A \text{ and } B) = 0.2 \).

### (b) Are \( A \) and \( B \) mutually exclusive?

Two events are mutually exclusive if they cannot occur at the same time, i.e., \( P(A \cap B) = 0 \).

Since \( P(A \cap B) = 0.2 \neq 0 \), \( A \) and \( B \) are **not** mutually exclusive.

### (c) Are \( A \) and \( B \) independent?

Two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \times P(B) \).

Calculating \( P(A) \times P(B) \):

\[
P(A) \times P(B) = 0.6 \times 0.4 = 0.24
\]

Since \( P(A \cap B) = 0.2 \neq 0.24 \), \( A \) and \( B \) are **not** independent.

---

Would you like more details or have any specific questions on these calculations?

### Related Questions:
1. How do you calculate the probability of the complement of \( A \) and \( B \)?
2. What is the probability of either \( A \) or \( B \) occurring, given their independence?
3. How would the result change if \( P(A) \) was greater than 1?
4. What does it mean for events to be independent in real-life scenarios?
5. How do you verify mutual exclusivity if the events are part of a complex system?

**Tip:** Always remember that mutually exclusive events cannot happen simultaneously, while independent events have no effect on each other's occurrence.

Let A and B be events with P(A) = 0.6, P(B) = 0.4, and P(A or B) = 0.8. (a) Compute P(A and B). (b) Are A and B mutually exclusive? Explain. (c) Are A and B independent? Explain.

Learn how to calculate the probability of events A and B occurring, determine if they are mutually exclusive, and check for their independence. This problem uses key probability concepts, including event union, mutual exclusivity, and independence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation