If A and B are two mutually exclusive and exhaustive events with P(B) = 3P (A), then what is the value of `P (barB)`?

Options

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

Answer

To solve the problem, we need to find the value of \( P(\bar{B}) \) given that \( A \) and \( B \) are mutually exclusive and exhaustive events, and \( P(B) = 3P(A) \).

### Step-by-Step Solution:

1. **Understanding the Terms**:
- Mutually exclusive events mean that \( A \) and \( B \) cannot occur at the same time. Thus, \( P(A \cap B) = 0 \).
- Exhaustive events mean that at least one of the events must occur. Thus, \( P(A) + P(B) = 1 \).

2. **Setting Up the Equations**:
- We are given that \( P(B) = 3P(A) \).
- Let \( P(A) = x \). Then, \( P(B) = 3x \).

3. **Using the Exhaustive Property**:
- Since \( A \) and \( B \) are exhaustive, we can write:
\[ P(A) + P(B) = 1 \]
- Substituting the values we have:
\[ x + 3x = 1 \]

4. **Solving for \( x \)**:
- Combine like terms:
\[ 4x = 1 \]
- Now, divide both sides by 4:
\[ x = \frac{1}{4} \]
- So, \( P(A) = \frac{1}{4} \).

5. **Finding \( P(B) \)**:
- Now substitute back to find \( P(B) \):
\[ P(B) = 3P(A) = 3 \times \frac{1}{4} = \frac{3}{4} \]

6. **Finding \( P(\bar{B}) \)**:
- The probability of the complement of \( B \) is given by:
\[ P(\bar{B}) = 1 - P(B) \]
- Substitute \( P(B) \):
\[ P(\bar{B}) = 1 - \frac{3}{4} = \frac{1}{4} \]

### Final Answer:
Thus, the value of \( P(\bar{B}) \) is \( \frac{1}{4} \).