If P(A)=0.4 P(B) = 0.8 and `P (B/A) = 0.6 ` then find `P(A cup B).`

Answer

To find \( P(A \cup B) \), we can use the formula:

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

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = 0.4 \)
- \( P(B) = 0.8 \)
- \( P(B|A) = 0.6 \)

### Step 2: Find \( P(A \cap B) \)
Using the definition of conditional probability:

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

We can rearrange this to find \( P(A \cap B) \):

\[ P(A \cap B) = P(B|A) \times P(A) \]

Substituting the known values:

\[ P(A \cap B) = 0.6 \times 0.4 = 0.24 \]

### Step 3: Substitute values into the formula for \( P(A \cup B) \)
Now we can substitute \( P(A) \), \( P(B) \), and \( P(A \cap B) \) into the formula:

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

This gives us:

\[ P(A \cup B) = 0.4 + 0.8 - 0.24 \]

### Step 4: Calculate \( P(A \cup B) \)
Now perform the calculation:

\[ P(A \cup B) = 1.2 - 0.24 = 0.96 \]

### Final Answer
Thus, the probability \( P(A \cup B) \) is:

\[ P(A \cup B) = 0.96 \]

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