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

Explanation

To solve the problem step by step, we will use the provided probabilities and the relationships between them.

### Step 1: Identify the given probabilities
We have the following probabilities given in the problem:
- \( P(A \cup B) = 0.5 \)
- \( P(\bar{B}) = 0.8 \)
- \( P(A|B) = 0.4 \)

### Step 2: Calculate \( P(B) \)
We know that the probability of an event and its complement add up to 1. Therefore, we can find \( P(B) \) as follows:
\[ P(B) = 1 - P(\bar{B}) = 1 - 0.8 = 0.2 \]

### Step 3: Use the conditional probability to find \( P(A \cap B) \)
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 \( P(A \cap B) \):
\[ P(A \cap B) = P(A|B) \times P(B) \]
Substituting the values we have:
\[ P(A \cap B) = 0.4 \times 0.2 = 0.08 \]

### Step 4: Conclusion
Thus, the probability \( P(A \cap B) \) is equal to \( 0.08 \).

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