If P(A) = 0.4, P(B) = x, `P(A uu B)` = 0. 7 and the events A and B are independent, then x =

Options

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

Answer

To solve the problem, we will use the formula for the probability of the union of two independent events.

Given:
- \( P(A) = 0.4 \)
- \( P(B) = x \)
- \( P(A \cup B) = 0.7 \)

Since A and B are independent events, we know that:
\[ P(A \cap B) = P(A) \cdot P(B) = P(A) \cdot x = 0.4x \]

The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the known values into the formula:
\[ 0.7 = 0.4 + x - 0.4x \]

Now, we will simplify this equation step by step:

1. Combine like terms:
\[ 0.7 = 0.4 + x - 0.4x \]
\[ 0.7 = 0.4 + (1 - 0.4)x \]
\[ 0.7 = 0.4 + 0.6x \]

2. Subtract 0.4 from both sides:
\[ 0.7 - 0.4 = 0.6x \]
\[ 0.3 = 0.6x \]

3. Divide both sides by 0.6 to solve for x:
\[ x = \frac{0.3}{0.6} \]
\[ x = 0.5 \]

Thus, the value of \( x \) is \( 0.5 \).

### Summary of Steps:
1. Write down the known probabilities.
2. Use the formula for the union of independent events.
3. Substitute the known values into the formula.
4. Simplify the equation.
5. Solve for \( x \).