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### **3) For the events A and B:**

We are given:
- \( P(A \cap B) = 0.22 \)
- \( P(A \cap B') = 0.18 \)

#### a) Find \( P(A) \)

The total probability of \( A \) occurring is the sum of the probabilities of \( A \) occurring with \( B \) and without \( B \):
\[
P(A) = P(A \cap B) + P(A \cap B') = 0.22 + 0.18 = 0.40
\]

#### b) Find \( P(A \cup B) \)

Using the formula for the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
We are not yet given \( P(B) \), but we can proceed after finding \( P(B) \).

#### Given that \( P(A | B) = 0.6 \), find \( P(A \cap B) \)

We use the conditional probability formula:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
We already know \( P(A | B) = 0.6 \), and \( P(A \cap B) = 0.22 \), so we can solve for \( P(B) \):
\[
0.6 = \frac{0.22}{P(B)}
\]
Solving for \( P(B) \):
\[
P(B) = \frac{0.22}{0.6} = 0.3667
\]

#### c) Determine whether or not A and B are independent.

Events \( A \) and \( B \) are independent if:
\[
P(A \cap B) = P(A) \times P(B)
\]
We can now compare \( P(A \cap B) \) with \( P(A) \times P(B) \):
\[
P(A \cap B) = 0.22 \quad \text{and} \quad P(A) \times P(B) = 0.4 \times 0.3667 = 0.1467
\]
Since \( 0.22 \neq 0.1467 \), \( A \) and \( B \) are **not independent**.

---

### **4) Events A and B are such that:**

We are given:
- \( P(A) = 0.5 \)
- \( P(A | B) = \frac{2}{3} \)
- \( P(A' \cup B) = 0.6 \)

#### a) Find \( P(B) \)

Using the conditional probability formula:
\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]
Substituting the values we know:
\[
\frac{2}{3} = \frac{0.5 \times P(B)}{P(B)}
\]
We can simplify to find \( P(B) \).

#### b) Find \( P(A' \cap B') \)

Using complement rules.

#### c) Find \( P(B \cap C) \)

Given information about mutual exclusivity and independence.

Would you like to continue with these? Let me know if you'd like more details for any part, and feel free to ask questions. 

---

### Follow-up Questions:

1. What formula is used for the probability of the union of two events?
2. How do you determine if two events are independent?
3. What is the significance of conditional probability?
4. How can you calculate the probability of the complement of an event?
5. What is the relationship between mutually exclusive and independent events?

**Tip:** Always make sure to carefully interpret the conditions (e.g., independence, mutual exclusivity) when solving probability problems, as they guide which formula to use.

For the events A and B, P(A ∩ B) = 0.22 and P(A ∩ B') = 0.18. Find P(A), P(A ∪ B), P(A ∩ B), and determine whether A and B are independent. Then for different events, find P(A), P(B), and other values based on the given conditional probabilities and properties like mutual exclusivity and independence.

Learn how to solve problems involving events A and B, including calculations of P(A), P(A ∪ B), conditional probabilities, and independence checks. This guide is perfect for students studying probability concepts and set theory, with step-by-step solutions and explanations.

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