If `P(AnnB)=1/2,P( A nn B )=1/3, P(A)=p , P(B)=2p ,` then find the value of `pdot`

Explanation

To solve the problem, we will use the given probabilities and the properties of probability.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have \( P(A \cap B) = \frac{1}{2} \)
- We have \( P(A' \cap B') = \frac{1}{3} \) (where \( A' \) is the complement of \( A \) and \( B' \) is the complement of \( B \))
- We have \( P(A) = p \) and \( P(B) = 2p \)

2. **Use the Complement Rule**:
- From the property of complements, we know that:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
- Therefore, we can express \( P(A \cup B) \) as:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - \frac{1}{3} = \frac{2}{3} \]

3. **Use the Formula for Union of Two Events**:
- The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the known values:
\[ P(A \cup B) = p + 2p - \frac{1}{2} \]
- This simplifies to:
\[ P(A \cup B) = 3p - \frac{1}{2} \]

4. **Set the Two Expressions for \( P(A \cup B) \) Equal**:
- We have two expressions for \( P(A \cup B) \):
\[ 3p - \frac{1}{2} = \frac{2}{3} \]

5. **Solve for \( p \)**:
- Rearranging the equation:
\[ 3p = \frac{2}{3} + \frac{1}{2} \]
- To add the fractions, find a common denominator (which is 6):
\[ 3p = \frac{4}{6} + \frac{3}{6} = \frac{7}{6} \]
- Now, divide both sides by 3:
\[ p = \frac{7}{6} \cdot \frac{1}{3} = \frac{7}{18} \]

6. **Final Answer**:
- Therefore, the value of \( p \) is:
\[ p = \frac{7}{18} \]