If `P(A)= (2)/(3), P(B)= (2)/(5)` and `P( A uu B)- P(A nnB)= (2)/(5)` then what is `P(A nn B)` equal to ?

Options

A. `3//5`
B. `5//11`
C. `1//3`
D. `3//11`

Answer

To solve the problem, we need to find \( P(A \cap B) \) given the following information:

- \( P(A) = \frac{2}{3} \)
- \( P(B) = \frac{2}{5} \)
- \( P(A \cup B) - P(A \cap B) = \frac{2}{5} \)

### Step-by-step solution:

1. **Identify the formulas**:
We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

2. **Assign variables**:
Let:
- \( P(A \cup B) = X \)
- \( P(A \cap B) = Y \)

Then we can rewrite the equation as:
\[ X - Y = \frac{2}{5} \]

3. **Substitute known values**:
Substitute \( P(A) \) and \( P(B) \) into the union formula:
\[ X = P(A) + P(B) - Y = \frac{2}{3} + \frac{2}{5} - Y \]

4. **Find a common denominator**:
The least common multiple of 3 and 5 is 15. Convert the fractions:
\[ P(A) = \frac{2}{3} = \frac{10}{15}, \quad P(B) = \frac{2}{5} = \frac{6}{15} \]
Thus,
\[ X = \frac{10}{15} + \frac{6}{15} - Y = \frac{16}{15} - Y \]

5. **Substitute into the equation**:
Now substitute \( X \) back into the equation \( X - Y = \frac{2}{5} \):
\[ \left(\frac{16}{15} - Y\right) - Y = \frac{2}{5} \]
Simplifying gives:
\[ \frac{16}{15} - 2Y = \frac{2}{5} \]

6. **Find a common denominator for the fractions**:
The least common multiple of 15 and 5 is 15. Convert \( \frac{2}{5} \):
\[ \frac{2}{5} = \frac{6}{15} \]
Thus, the equation becomes:
\[ \frac{16}{15} - 2Y = \frac{6}{15} \]

7. **Rearranging the equation**:
Rearranging gives:
\[ 2Y = \frac{16}{15} - \frac{6}{15} \]
\[ 2Y = \frac{10}{15} \]

8. **Solve for \( Y \)**:
Dividing both sides by 2:
\[ Y = \frac{10}{15} \div 2 = \frac{10}{30} = \frac{1}{3} \]

9. **Conclusion**:
Therefore, the probability \( P(A \cap B) \) is:
\[ P(A \cap B) = \frac{1}{3} \]