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

Options

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

Answer

To solve the problem, we need to find \( P(A) \) using the given probabilities. We are provided with the following information:

- \( P(B) = \frac{3}{5} \)
- \( P(A|B) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{4}{5} \)

### Step 1: Find \( P(A \cap B) \)

Using the definition of conditional probability, we know that:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

From this, we can rearrange the formula to find \( P(A \cap B) \):

\[ P(A \cap B) = P(A|B) \cdot P(B) \]

Substituting the values we have:

\[ P(A \cap B) = \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10} \]

### Step 2: Use the formula for \( P(A \cup B) \)

We know that:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

We can rearrange this to find \( P(A) \):

\[ P(A) = P(A \cup B) + P(A \cap B) - P(B) \]

### Step 3: Substitute the known values

Now we substitute the values we have into the equation:

\[ P(A) = \frac{4}{5} + \frac{3}{10} - \frac{3}{5} \]

### Step 4: Find a common denominator

To perform the addition and subtraction, we need a common denominator. The least common multiple of 5 and 10 is 10. We convert all fractions:

\[ P(A) = \frac{4}{5} = \frac{8}{10}, \quad P(B) = \frac{3}{5} = \frac{6}{10} \]

Now substituting these values back into the equation:

\[ P(A) = \frac{8}{10} + \frac{3}{10} - \frac{6}{10} \]

### Step 5: Perform the arithmetic

Now we can simplify:

\[ P(A) = \frac{8 + 3 - 6}{10} = \frac{5}{10} = \frac{1}{2} \]

### Conclusion

Thus, the probability \( P(A) \) is:

\[ P(A) = \frac{1}{2} \]

### Answer

The answer is \( \frac{1}{2} \), which corresponds to option (a).

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