If A and B are events such that `P(A uu B) = 0.5, P(barB)= 0.8` and `P(A//B)= 0.4` then what is `P(A nn B)` equal to:

Options

A. 0.08
B. 0.02
C. 0.8
D. 0.2

Answer

To solve the problem step by step, we will use the given probabilities and the relevant formulas in probability theory.

### Step 1: Identify the given probabilities
We have the following probabilities:
- \( P(A \cup B) = 0.5 \) (Probability of A union B)
- \( P(\bar{B}) = 0.8 \) (Probability of not B)
- \( P(A|B) = 0.4 \) (Conditional probability of A given B)

### Step 2: Calculate \( P(B) \)
Since we know \( P(\bar{B}) \), we can find \( P(B) \) using the formula:
\[ P(B) = 1 - P(\bar{B}) \]
Substituting the value:
\[ P(B) = 1 - 0.8 = 0.2 \]

### Step 3: Use the conditional probability formula
The conditional probability \( P(A|B) \) is defined as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
From this, we can rearrange the formula to find \( P(A \cap B) \):
\[ P(A \cap B) = P(A|B) \times P(B) \]

### Step 4: Substitute the known values
Now we can substitute the values we have:
\[ P(A \cap B) = P(A|B) \times P(B) = 0.4 \times 0.2 \]

### Step 5: Calculate \( P(A \cap B) \)
Now, performing the multiplication:
\[ P(A \cap B) = 0.4 \times 0.2 = 0.08 \]

### Conclusion
Thus, the probability \( P(A \cap B) \) is equal to \( 0.08 \).

### Final Answer
\[ P(A \cap B) = 0.08 \]