If A and B are events sucn that `P(A) =0.3 P(B)=0.2 " and " P(A nn B) =0.1` then `P(bar(A) nn B)=?`

Options

A. `0.2`
B. `0.1`
C. `0.4`
D. `0.5`

Explanation

To solve the problem, we need to find \( P(\bar{A} \cap B) \), which is the probability that event B occurs while event A does not occur. We can use the formula for the intersection of events and the properties of probability.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- \( P(A) = 0.3 \)
- \( P(B) = 0.2 \)
- \( P(A \cap B) = 0.1 \)

2. **Use the Formula for the Probability of the Complement**:
We know that:
\[ P(B) = P(A \cap B) + P(\bar{A} \cap B) \]
This means that the probability of B can be expressed as the sum of the probability of A and B occurring together and the probability of B occurring without A.

3. **Rearrange the Formula**:
To find \( P(\bar{A} \cap B) \), we can rearrange the formula:
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]

4. **Substitute the Known Values**:
Now, substitute the values we have:
\[ P(\bar{A} \cap B) = 0.2 - 0.1 \]

5. **Calculate the Result**:
\[ P(\bar{A} \cap B) = 0.1 \]

### Final Answer:
Thus, the probability \( P(\bar{A} \cap B) = 0.1 \).

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