If `P(A cup B) =1/2 and P(A cap B) = 1/3 ,P(A) = p ` and P(B) = 2p , then the value of p

Options

A. `1/9`
B. `1/3`
C. `4/9`
D. `5/18`

Answer

To solve the problem, we will use the formula for the probability of the union of two events, which is given by:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Given:
- \( P(A \cup B) = \frac{1}{2} \)
- \( P(A \cap B) = \frac{1}{3} \)
- \( P(A) = p \)
- \( P(B) = 2p \)

Now, we can substitute the known values into the formula:

1. Substitute the values into the union formula:
\[ \frac{1}{2} = p + 2p - \frac{1}{3} \]

2. Combine like terms:
\[ \frac{1}{2} = 3p - \frac{1}{3} \]

3. To eliminate the fractions, find a common denominator. The least common multiple of 2 and 3 is 6. Multiply the entire equation by 6:
\[ 6 \cdot \frac{1}{2} = 6 \cdot 3p - 6 \cdot \frac{1}{3} \]
This simplifies to:
\[ 3 = 18p - 2 \]

4. Now, add 2 to both sides:
\[ 3 + 2 = 18p \]
\[ 5 = 18p \]

5. Finally, solve for \( p \) by dividing both sides by 18:
\[ p = \frac{5}{18} \]

Thus, the value of \( p \) is \( \frac{5}{18} \).