If `P(AcupB)=0*9,P(B)=0*4` when A and B are independent events then P(A)=_____

Answer

To solve the problem, we need to find \( P(A) \) given that \( P(A \cup B) = 0.9 \) and \( P(B) = 0.4 \), with the events A and B being independent.

### Step-by-Step Solution:

1. **Understand the formula for the union of two independent events**:
The formula for the probability of the union of two events A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

2. **Substitute the known values into the formula**:
We know that \( P(A \cup B) = 0.9 \) and \( P(B) = 0.4 \). Substituting these values into the formula gives:
\[ 0.9 = P(A) + 0.4 - P(A) \cdot 0.4 \]

3. **Rearrange the equation**:
To isolate \( P(A) \), we can rearrange the equation:
\[ 0.9 = P(A) + 0.4 - 0.4 P(A) \]
Combine the terms involving \( P(A) \):
\[ 0.9 = P(A)(1 - 0.4) + 0.4 \]
\[ 0.9 = P(A)(0.6) + 0.4 \]

4. **Subtract 0.4 from both sides**:
\[ 0.9 - 0.4 = P(A)(0.6) \]
\[ 0.5 = P(A)(0.6) \]

5. **Divide both sides by 0.6**:
To find \( P(A) \), divide both sides by 0.6:
\[ P(A) = \frac{0.5}{0.6} \]

6. **Simplify the fraction**:
\[ P(A) = \frac{5}{6} \]

Thus, the final answer is:
\[ P(A) = \frac{5}{6} \]