If P(A) = 0.7, P(B) = 0.4, `P(A nn B)` = 0.3, then `P(A nn B') = `

Options

A. `(2)/(5)`
B. 1
C. `0.7`
D. `0.42`

Answer

To solve the problem, we need to find \( P(A \cap B') \), where:

- \( P(A) = 0.7 \)
- \( P(B) = 0.4 \)
- \( P(A \cap B) = 0.3 \)

### Step-by-Step Solution:

1. **Understand the Complement**:
We know that \( B' \) (B complement) represents the event that B does not occur. Therefore, \( P(A \cap B') \) represents the probability that A occurs while B does not.

2. **Use the Formula**:
We can use the relationship:
\[ P(A) = P(A \cap B) + P(A \cap B') \]
This means that the total probability of A can be split into two parts: the probability of A occurring with B and the probability of A occurring without B.

3. **Substitute Known Values**:
We already have \( P(A) = 0.7 \) and \( P(A \cap B) = 0.3 \). We can substitute these values into the formula:
\[ 0.7 = 0.3 + P(A \cap B') \]

4. **Solve for \( P(A \cap B') \)**:
Rearranging the equation gives us:
\[ P(A \cap B') = 0.7 - 0.3 \]
\[ P(A \cap B') = 0.4 \]

5. **Convert to Fraction (Optional)**:
We can express \( 0.4 \) as a fraction:
\[ 0.4 = \frac{4}{10} = \frac{2}{5} \]

Thus, the final answer is:
\[ P(A \cap B') = 0.4 \text{ or } \frac{2}{5} \]