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 find the probability that event A occurs but event B does not occur, we can use the following steps:

### Step-by-Step Solution:

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

2. **Understand the Required Probability:**
- We need to find the probability that A happens but B does not happen, which is represented as \( P(A \cap B') \) where \( B' \) is the complement of B.

3. **Use the Formula for Conditional Probability:**
- We can express \( P(A \cap B') \) using the relationship:
\[ P(A) = P(A \cap B) + P(A \cap B') \]
- Rearranging this gives us:
\[ P(A \cap B') = P(A) - P(A \cap B) \]

4. **Substitute the Known Values:**
- Now we can substitute the values we have:
\[ P(A \cap B') = P(A) - P(A \cap B) = \frac{1}{2} - \frac{1}{3} \]

5. **Calculate the Result:**
- To perform the subtraction, we need a common denominator. The least common multiple of 2 and 3 is 6.
- Convert \( \frac{1}{2} \) and \( \frac{1}{3} \) to have a denominator of 6:
\[ P(A) = \frac{1}{2} = \frac{3}{6} \]
\[ P(A \cap B) = \frac{1}{3} = \frac{2}{6} \]
- Now substitute these values back into the equation:
\[ P(A \cap B') = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

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