If A and B two events such that P(A)=0.2, P(B)=0.4 and `P(A cup B)=0.5`, then value of `P(A//B)` is ?

Options

A. 0.1
B. 0.25
C. 0.5
D. 0.08

Explanation

To find the value of \( P(A|B) \), we can use the formula for conditional probability:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = 0.2 \)
- \( P(B) = 0.4 \)
- \( P(A \cup B) = 0.5 \)

### Step 2: Use the formula for the union of two events
We can express \( P(A \cup B) \) using the formula:

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

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:

\[ 0.5 = 0.2 + 0.4 - P(A \cap B) \]

### Step 4: Simplify the equation
Now, simplify the equation:

\[ 0.5 = 0.6 - P(A \cap B) \]

### Step 5: Solve for \( P(A \cap B) \)
Rearranging the equation gives:

\[ P(A \cap B) = 0.6 - 0.5 = 0.1 \]

### Step 6: Use the value of \( P(A \cap B) \) to find \( P(A|B) \)
Now, we can substitute \( P(A \cap B) \) back into the conditional probability formula:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.1}{0.4} \]

### Step 7: Simplify the fraction
This simplifies to:

\[ P(A|B) = \frac{1}{4} = 0.25 \]

### Final Answer
Thus, the value of \( P(A|B) \) is \( 0.25 \).

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