If A and B are any two events such that `(A uu B) = (3)/(4), P(A nn B) = (1)/(4)` and `P(barA)= (2)/(3)` where `barA` stands for the complementary event of A. then what is P(B)?

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `(1)/(9)`
D. `(2)/(9)`

Answer

To find \( P(B) \) given the probabilities of events \( A \) and \( B \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- \( P(A \cup B) = \frac{3}{4} \)
- \( P(A \cap B) = \frac{1}{4} \)
- \( P(\bar{A}) = \frac{2}{3} \)

2. **Calculate \( P(A) \)**:
- Since \( P(\bar{A}) = 1 - P(A) \), we can find \( P(A) \):
\[ P(A) = 1 - P(\bar{A}) = 1 - \frac{2}{3} = \frac{1}{3} \]

3. **Use the Formula for Union of Two Events**:
- The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the known values:
\[ \frac{3}{4} = P(A) + P(B) - P(A \cap B) \]
\[ \frac{3}{4} = \frac{1}{3} + P(B) - \frac{1}{4} \]

4. **Rearranging the Equation**:
- First, we need to combine the terms on the right side:
\[ P(B) = \frac{3}{4} + \frac{1}{4} - \frac{1}{3} \]
- Combine \( \frac{3}{4} \) and \( \frac{1}{4} \):
\[ P(B) = 1 - \frac{1}{3} \]

5. **Finding a Common Denominator**:
- To subtract \( \frac{1}{3} \) from 1, convert 1 into a fraction with a denominator of 3:
\[ 1 = \frac{3}{3} \]
- Now, perform the subtraction:
\[ P(B) = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \]

6. **Final Result**:
- Therefore, the probability of event \( B \) is:
\[ P(B) = \frac{2}{3} \]

### Summary:
The probability \( P(B) \) is \( \frac{2}{3} \).