If A and B are two events such that P(B) = `3/5` , P(A/B) =`1/2 and P(A cup B) = 4/5` then P(A) equals

Options

A. `3/10`
B. `1/5`
C. `1/2`
D. `3/5`

Answer

To solve the problem, we need to find the probability of event A, given the following information:

- P(B) = 3/5
- P(A/B) = 1/2
- P(A ∪ B) = 4/5

### Step-by-Step Solution:

1. **Understanding Conditional Probability**:
We know that:
\[ P(A/B) = \frac{P(A \cap B)}{P(B)} \]
From this, we can express \( P(A \cap B) \):
\[ P(A \cap B) = P(A/B) \cdot P(B) \]

2. **Substituting the Known Values**:
Substitute the known values into the equation:
\[ P(A \cap B) = \frac{1}{2} \cdot \frac{3}{5} \]
This simplifies to:
\[ P(A \cap B) = \frac{3}{10} \]

3. **Using the Union Formula**:
We use the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Rearranging this gives us:
\[ P(A) = P(A \cup B) - P(B) + P(A \cap B) \]

4. **Substituting the Known Values Again**:
Now substitute the known values into this equation:
\[ P(A) = \frac{4}{5} - \frac{3}{5} + \frac{3}{10} \]

5. **Finding a Common Denominator**:
The common denominator for the fractions is 10. Convert each term:
\[ P(A) = \frac{8}{10} - \frac{6}{10} + \frac{3}{10} \]

6. **Performing the Arithmetic**:
Now combine the fractions:
\[ P(A) = \frac{8 - 6 + 3}{10} = \frac{5}{10} \]

7. **Simplifying the Result**:
Finally, simplify:
\[ P(A) = \frac{1}{2} \]

### Final Answer:
Thus, the probability of event A is:
\[ P(A) = \frac{1}{2} \]