Two events A and B are such that P(not B) = 0.8, `P (A uu B) = 0.5` and `P(A|B) = 0.4`. Then P(A) is equal to:

Options

A. 0.28
B. 0.32
C. 0.38
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate P(B)
We know that:
\[ P(\text{not } B) = 0.8 \]
To find \( P(B) \):
\[ P(B) = 1 - P(\text{not } B) \]
\[ P(B) = 1 - 0.8 = 0.2 \]

### Step 2: Use the Conditional Probability Formula
We are given:
\[ P(A|B) = 0.4 \]
The formula for conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the known values:
\[ 0.4 = \frac{P(A \cap B)}{0.2} \]
To find \( P(A \cap B) \):
\[ P(A \cap B) = 0.4 \times 0.2 = 0.08 \]

### Step 3: Use the Formula for P(A ∪ B)
We are given:
\[ P(A \cup B) = 0.5 \]
The formula for the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ 0.5 = P(A) + 0.2 - 0.08 \]

### Step 4: Solve for P(A)
Rearranging the equation:
\[ P(A) = 0.5 - 0.2 + 0.08 \]
\[ P(A) = 0.5 - 0.2 + 0.08 = 0.3 + 0.08 = 0.38 \]

Thus, the probability \( P(A) \) is:
\[ P(A) = 0.38 \]

### Final Answer
The probability of event A is \( P(A) = 0.38 \).

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