Events A and B are given such that P(A) = 1/3, P(A or B ) = 4/9 and P(B)=2/9. Show that A and B are neither independent nor mutually exclusive.

To show that events \( A \) and \( B \) are neither independent nor mutually exclusive, we'll begin by using the given probabilities:

- \( P(A) = \frac{1}{3} \)
- \( P(B) = \frac{2}{9} \)
- \( P(A \cup B) = \frac{4}{9} \)

### Step 1: Check if \( A \) and \( B \) are mutually exclusive

Events \( A \) and \( B \) are mutually exclusive if they cannot happen together, i.e., \( P(A \cap B) = 0 \). 

Using the formula for the probability of the union of two events:

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

Substitute the known values:

\[
\frac{4}{9} = \frac{1}{3} + \frac{2}{9} - P(A \cap B)
\]

Convert \( \frac{1}{3} \) to \( \frac{3}{9} \):

\[
\frac{4}{9} = \frac{3}{9} + \frac{2}{9} - P(A \cap B)
\]

Simplify:

\[
\frac{4}{9} = \frac{5}{9} - P(A \cap B)
\]

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

\[
P(A \cap B) = \frac{5}{9} - \frac{4}{9} = \frac{1}{9}
\]

Since \( P(A \cap B) = \frac{1}{9} \), which is not 0, events \( A \) and \( B \) are **not mutually exclusive**.

### Step 2: Check if \( A \) and \( B \) are independent

Events \( A \) and \( B \) are independent if:

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

We already calculated \( P(A \cap B) = \frac{1}{9} \). Now check if this equals \( P(A) \cdot P(B) \).

Calculate \( P(A) \cdot P(B) \):

\[
P(A) \cdot P(B) = \frac{1}{3} \times \frac{2}{9} = \frac{2}{27}
\]

Clearly, \( P(A \cap B) = \frac{1}{9} \neq \frac{2}{27} \), so events \( A \) and \( B \) are **not independent**.

### Conclusion:
- Events \( A \) and \( B \) are **not mutually exclusive** because \( P(A \cap B) \neq 0 \).
- Events \( A \) and \( B \) are **not independent** because \( P(A \cap B) \neq P(A) \cdot P(B) \).

Would you like further clarification on any part of this?

Here are five related questions to explore:
1. How would the situation change if \( P(A \cap B) = 0 \)?
2. What does mutual exclusivity imply about real-world events?
3. Can events be independent but not mutually exclusive?
4. How is the complement of an event related to its probability?
5. How would we approach this problem if \( P(A) \) and \( P(B) \) were given in percentages?

**Tip:** Always remember the key formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) for combining event probabilities.

Explore how to determine that events A and B are neither independent nor mutually exclusive, using the probabilities P(A) = 1/3, P(B) = 2/9, and P(A or B) = 4/9. This problem involves key probability concepts and theorems like the union and intersection of events. Suitable for students in Grades 10-12.

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