If A and B are mutually exclusive and exhaustive events and P(A) = `1/3` P(B), then P(A) is equal to (i) `1/4` (ii) `3/4` (iii) `1/2` (iv) `3/8`

Options

A. `1/4`
B. `3/4`
C. `1/2`
D. `3/8`

Answer

To solve the problem, we need to find the probability of event A (P(A)) given that events A and B are mutually exclusive and exhaustive, and that P(A) = (1/3)P(B).

### Step-by-Step Solution:

1. **Understanding Mutual Exclusivity and Exhaustiveness**:
- Since A and B are mutually exclusive, it means that they cannot occur at the same time. Thus, P(A ∩ B) = 0.
- Since they are exhaustive, it means that at least one of them must occur. Therefore, P(A ∪ B) = 1.

2. **Setting Up the Equation**:
- We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Since P(A ∩ B) = 0 (mutually exclusive), we can simplify this to:
\[ P(A) + P(B) = 1 \]

3. **Expressing P(B) in Terms of P(A)**:
- From the problem statement, we have:
\[ P(A) = \frac{1}{3} P(B) \]
- Let P(A) = x. Then, substituting this into the equation gives:
\[ x + P(B) = 1 \]
- We can express P(B) in terms of x:
\[ P(B) = 1 - x \]

4. **Substituting P(B) into the Equation**:
- Now, substitute P(B) into the equation derived from the relationship between P(A) and P(B):
\[ x = \frac{1}{3}(1 - x) \]

5. **Solving for x**:
- Multiply both sides by 3 to eliminate the fraction:
\[ 3x = 1 - x \]
- Now, add x to both sides:
\[ 3x + x = 1 \]
\[ 4x = 1 \]
- Finally, divide by 4:
\[ x = \frac{1}{4} \]

6. **Conclusion**:
- Therefore, P(A) = x = \( \frac{1}{4} \).

### Final Answer:
P(A) is equal to \( \frac{1}{4} \).