If A and B are independent events such that `P(A) =0.4 ,P(B) =x` and
`P(A uu B) =0.5 "then " x=?`

Options

A. `(4)/(5)`
B. `0.1`
C. `(1)/(6)`
D. None of these

Explanation

To solve the problem, we need to find the value of \( x \) given the probabilities of independent events \( A \) and \( B \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that \( P(A) = 0.4 \)
- We know that \( P(B) = x \)
- We know that \( P(A \cup B) = 0.5 \)

2. **Use the Formula for Union of Two Events**:
The formula for the probability of the union of two independent events \( A \) and \( B \) is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

3. **Calculate \( P(A \cap B) \)**:
Since \( A \) and \( B \) are independent, we can use the property:
\[ P(A \cap B) = P(A) \cdot P(B) = P(A) \cdot x \]
Substituting the value of \( P(A) \):
\[ P(A \cap B) = 0.4 \cdot x \]

4. **Substitute into the Union Formula**:
Now we can substitute \( P(A) \), \( P(B) \), and \( P(A \cap B) \) into the union formula:
\[ 0.5 = P(A) + P(B) - P(A \cap B) \]
This becomes:
\[ 0.5 = 0.4 + x - (0.4 \cdot x) \]

5. **Simplify the Equation**:
Rearranging the equation gives:
\[ 0.5 = 0.4 + x - 0.4x \]
Combine like terms:
\[ 0.5 = 0.4 + (1 - 0.4)x \]
\[ 0.5 = 0.4 + 0.6x \]

6. **Isolate \( x \)**:
Subtract \( 0.4 \) from both sides:
\[ 0.5 - 0.4 = 0.6x \]
\[ 0.1 = 0.6x \]

7. **Solve for \( x \)**:
Divide both sides by \( 0.6 \):
\[ x = \frac{0.1}{0.6} = \frac{1}{6} \]

### Final Answer:
Thus, the value of \( x \) is \( \frac{1}{6} \).