If `P(A) = 1/2` and `P(AcapB)=1/3`, then the probability that A happens but B does not happen is

Options

A. `1/2`
B. `1/3`
C. `5/6`
D. `1/6`

Answer

To solve the problem, we need to find the probability that event A occurs but event B does not occur, denoted as \( P(A \cap B') \).

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We are given \( P(A) = \frac{1}{2} \)
- We are also given \( P(A \cap B) = \frac{1}{3} \)

2. **Use the Relationship Between Events:**
- The probability that A occurs but B does not occur can be expressed using the formula:
\[ P(A \cap B') = P(A) - P(A \cap B) \]
- Here, \( P(A \cap B') \) represents the probability of A occurring and B not occurring.

3. **Substitute the Given Values:**
- Now, we substitute the known probabilities into the equation:
\[ P(A \cap B') = P(A) - P(A \cap B) = \frac{1}{2} - \frac{1}{3} \]

4. **Calculate the Right Side:**
- To perform the subtraction, we need a common denominator. The least common multiple of 2 and 3 is 6.
- Rewrite the fractions:
\[ P(A) = \frac{1}{2} = \frac{3}{6} \]
\[ P(A \cap B) = \frac{1}{3} = \frac{2}{6} \]
- Now substitute these values:
\[ P(A \cap B') = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

5. **Final Result:**
- Therefore, the probability that A happens but B does not happen is:
\[ P(A \cap B') = \frac{1}{6} \]

### Summary:
The probability that A occurs but B does not occur is \( \frac{1}{6} \).