Let A & B be two events. Suppose `P(A) = 0.4` , `P(B) = p` and `P(AuuB)=0.7` The value of `p` for which `A` and `B` are independent is

Options

A. `1//3`
B. `1//4`
C. `1//2`
D. `1//5`

Explanation

To solve the problem, we need to find the value of \( p \) such that events \( A \) and \( B \) are independent given the probabilities \( P(A) = 0.4 \), \( P(B) = p \), and \( P(A \cup B) = 0.7 \).

### Step-by-Step Solution:

1. **Understanding Independence**:
For two events \( A \) and \( B \) to be independent, the following condition must hold:
\[ P(A \cap B) = P(A) \cdot P(B) \]

2. **Using the Formula for Union of Events**:
We know the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the independence condition into this formula gives:
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

3. **Substituting Known Values**:
We substitute the known values into the equation:
\[ 0.7 = 0.4 + p - (0.4 \cdot p) \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 0.7 = 0.4 + p - 0.4p \]
Combining like terms results in:
\[ 0.7 = 0.4 + p(1 - 0.4) \]
Simplifying further:
\[ 0.7 = 0.4 + 0.6p \]

5. **Isolating \( p \)**:
Subtract \( 0.4 \) from both sides:
\[ 0.7 - 0.4 = 0.6p \]
This simplifies to:
\[ 0.3 = 0.6p \]

6. **Solving for \( p \)**:
Divide both sides by \( 0.6 \):
\[ p = \frac{0.3}{0.6} = \frac{1}{2} \]

### Final Answer:
Thus, the value of \( p \) for which events \( A \) and \( B \) are independent is:
\[ p = \frac{1}{2} \]