If event A and B are mutually exclusive and exhaustive events and `P(A)=(1)/(3 )P(B)` then Find P (A)

Answer

To solve the problem step by step, we will use the information given about events A and B being mutually exclusive and exhaustive, along with the relationship between their probabilities.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- Mutually Exclusive: Events A and B cannot occur at the same time, which means \( P(A \cap B) = 0 \).
- Exhaustive: Events A and B cover all possible outcomes, so \( P(A \cup B) = 1 \).

2. **Setting Up the Relationship**:
- We are given that \( P(A) = \frac{1}{3} P(B) \). Let's denote \( P(B) \) as \( p \). Therefore, we can express \( P(A) \) as:
\[ P(A) = \frac{1}{3} p \]

3. **Using the Exhaustive Property**:
- Since A and B are exhaustive, we can write:
\[ P(A) + P(B) = 1 \]
- Substituting the expression for \( P(A) \):
\[ \frac{1}{3} p + p = 1 \]

4. **Combining Terms**:
- To combine the terms on the left side, we can express \( p \) as \( \frac{3}{3} p \):
\[ \frac{1}{3} p + \frac{3}{3} p = 1 \]
\[ \frac{4}{3} p = 1 \]

5. **Solving for \( P(B) \)**:
- To find \( p \), multiply both sides by \( \frac{3}{4} \):
\[ p = \frac{3}{4} \]
- Thus, \( P(B) = \frac{3}{4} \).

6. **Finding \( P(A) \)**:
- Now substitute \( p \) back into the equation for \( P(A) \):
\[ P(A) = \frac{1}{3} P(B) = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4} \]

### Final Answer:
\[ P(A) = \frac{1}{4} \]