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

Answer

To solve the problem, we need to find \( P(B) \) given that \( P(A) = 0.3 \) and \( P(A \cup B^c) = 0.8 \). Here are the steps to find \( P(B) \):

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that \( P(A) = 0.3 \).
- We also know that \( P(A \cup B^c) = 0.8 \).

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

3. **Express \( P(B^c) \)**:
Since \( P(B^c) = 1 - P(B) \), we can substitute this into the union formula:
\[ P(A \cup B^c) = P(A) + (1 - P(B)) - P(A \cap B^c) \]

4. **Calculate \( P(A \cap B^c) \)**:
Since \( A \) and \( B \) are independent events, we can express \( P(A \cap B^c) \) as:
\[ P(A \cap B^c) = P(A) \cdot P(B^c) = P(A) \cdot (1 - P(B)) \]
Substituting \( P(A) = 0.3 \):
\[ P(A \cap B^c) = 0.3 \cdot (1 - P(B)) \]

5. **Substitute Everything Back into the Union Formula**:
Now, substituting \( P(A) \) and \( P(A \cap B^c) \) back into the union formula:
\[ 0.8 = 0.3 + (1 - P(B)) - 0.3(1 - P(B)) \]

6. **Simplify the Equation**:
Simplifying the equation:
\[ 0.8 = 0.3 + 1 - P(B) - 0.3 + 0.3P(B) \]
This simplifies to:
\[ 0.8 = 1 - P(B) + 0.3P(B) \]
Rearranging gives:
\[ 0.8 = 1 - 0.7P(B) \]

7. **Solve for \( P(B) \)**:
Rearranging the equation to isolate \( P(B) \):
\[ 0.7P(B) = 1 - 0.8 \]
\[ 0.7P(B) = 0.2 \]
\[ P(B) = \frac{0.2}{0.7} = \frac{2}{7} \]

### Final Answer:
Thus, the probability \( P(B) \) is \( \frac{2}{7} \).

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