A and B are two events such that `P(A) = 0.3 and P (A uu B) = 0.8`. If A and B are independent, then P (B) is

Options

A. `2/3`
B. `3/8`
C. `2/7`
D. None of the above

Answer

To solve the problem, we need to find the probability of event B, given that events A and B are independent. We are provided with the following information:

- \( P(A) = 0.3 \)
- \( P(A \cup B) = 0.8 \)

Since A and B are independent, we can use the formula for the probability of the union of two independent events:

\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

### Step-by-step Solution:

1. **Write down the formula for the union of two independent events:**
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

2. **Substitute the known values into the formula:**
\[ 0.8 = 0.3 + P(B) - (0.3 \cdot P(B)) \]

3. **Rearrange the equation:**
\[ 0.8 = 0.3 + P(B) - 0.3P(B) \]
Combine the terms involving \( P(B) \):
\[ 0.8 = 0.3 + P(B)(1 - 0.3) \]
This simplifies to:
\[ 0.8 = 0.3 + 0.7P(B) \]

4. **Isolate \( P(B) \):**
Subtract 0.3 from both sides:
\[ 0.8 - 0.3 = 0.7P(B) \]
\[ 0.5 = 0.7P(B) \]

5. **Solve for \( P(B) \):**
Divide both sides by 0.7:
\[ P(B) = \frac{0.5}{0.7} = \frac{5}{7} \]

### Final Answer:
Thus, the probability of event B is:
\[ P(B) = \frac{5}{7} \]