If A and B are any two events having `P(A capB)=1/2` and `P(B) =2/3`,then find the `P(barA capB)`

Answer

To solve the problem, we need to find the probability of the event \( \bar{A} \cap B \), given that \( P(A \cap B) = \frac{1}{2} \) and \( P(B) = \frac{2}{3} \).

### Step-by-Step Solution:

1. **Understand the relationship between the probabilities**:
We know that:
\[ P(B) = P(A \cap B) + P(\bar{A} \cap B) \]
This means that the probability of event B can be divided into two parts: the part where A occurs (i.e., \( A \cap B \)) and the part where A does not occur (i.e., \( \bar{A} \cap B \)).

2. **Substitute the known values**:
We can substitute the known values into the equation:
\[ P(B) = \frac{2}{3}, \quad P(A \cap B) = \frac{1}{2} \]
Therefore, we can write:
\[ \frac{2}{3} = \frac{1}{2} + P(\bar{A} \cap B) \]

3. **Rearrange the equation to isolate \( P(\bar{A} \cap B) \)**:
To find \( P(\bar{A} \cap B) \), we can rearrange the equation:
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]
Substituting the values we have:
\[ P(\bar{A} \cap B) = \frac{2}{3} - \frac{1}{2} \]

4. **Find a common denominator to perform the subtraction**:
The common denominator for \( \frac{2}{3} \) and \( \frac{1}{2} \) is 6. We convert both fractions:
\[ \frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6} \]
Now we can subtract:
\[ P(\bar{A} \cap B) = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]

5. **Final answer**:
Therefore, the probability \( P(\bar{A} \cap B) \) is:
\[ P(\bar{A} \cap B) = \frac{1}{6} \]