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

Options

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

Explanation

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

### Step-by-Step Solution:

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

2. **Setting Up the Equations**:
- From the problem, we know that \( P(B) = 3P(A) \).
- Let \( P(A) = p \). Then, \( P(B) = 3p \).

3. **Using the Exhaustive Property**:
- Since A and B are exhaustive, we have:
\[ P(A) + P(B) = 1 \]
- Substituting the values we have:
\[ p + 3p = 1 \]
- This simplifies to:
\[ 4p = 1 \]

4. **Solving for \( P(A) \)**:
- Dividing both sides by 4 gives:
\[ p = \frac{1}{4} \]
- Thus, \( P(A) = \frac{1}{4} \).

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

6. **Calculating \( P(\bar{B}) \)**:
- The complement of event B, denoted as \( \bar{B} \), is given by:
\[ P(\bar{B}) = 1 - P(B) \]
- Substituting the value of \( P(B) \):
\[ P(\bar{B}) = 1 - \frac{3}{4} = \frac{1}{4} \]

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

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