If A and B be independent events with `P(A) = 1/4 and (P(A cupB) = 2P(B) - P(A) ` , then P(B) is equal to

Options

A. `2/5`
B. `1/5`
C. `3/5`
D. None of these

Answer

To solve the problem, we need to find the probability \( P(B) \) given the conditions of the events \( A \) and \( B \).

### Step-by-step Solution:

1. **Understand the Given Information:**
- We know that \( P(A) = \frac{1}{4} \).
- We also have the equation \( P(A \cup B) = 2P(B) - P(A) \).

2. **Use the Formula for Independent Events:**
- For independent events, the formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]

3. **Substitute the Known Values:**
- Substitute \( P(A) \) into the formula:
\[ P(A \cup B) = \frac{1}{4} + P(B) - \left(\frac{1}{4} \cdot P(B)\right) \]

4. **Set the Two Expressions for \( P(A \cup B) \) Equal:**
- We have two expressions for \( P(A \cup B) \):
\[ \frac{1}{4} + P(B) - \frac{1}{4}P(B) = 2P(B) - \frac{1}{4} \]

5. **Simplify the Left Side:**
- Combine the terms on the left side:
\[ \frac{1}{4} + P(B) - \frac{1}{4}P(B) = \frac{1}{4} + P(B)(1 - \frac{1}{4}) = \frac{1}{4} + \frac{3}{4}P(B) \]

6. **Set the Equation:**
- Now we set the left side equal to the right side:
\[ \frac{1}{4} + \frac{3}{4}P(B) = 2P(B) - \frac{1}{4} \]

7. **Rearranging the Equation:**
- Move all terms involving \( P(B) \) to one side and constant terms to the other:
\[ \frac{1}{4} + \frac{1}{4} = 2P(B) - \frac{3}{4}P(B) \]
\[ \frac{1}{2} = \frac{5}{4}P(B) \]

8. **Solve for \( P(B) \):**
- Multiply both sides by \( \frac{4}{5} \):
\[ P(B) = \frac{1/2 \cdot 4}{5} = \frac{2}{5} \]

### Final Answer:
Thus, \( P(B) = \frac{2}{5} \).

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