If A and B are independent events such that P(A)=p, P(B)=2p and P(Exactly one of A,B) `=5/9` , then find p.

Answer

To solve the problem, we need to find the value of \( p \) given the probabilities of independent events \( A \) and \( B \).

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have two independent events \( A \) and \( B \).
- The probabilities are given as \( P(A) = p \) and \( P(B) = 2p \).
- The probability of exactly one of the events occurring is given as \( P(\text{Exactly one of } A, B) = \frac{5}{9} \).

2. **Use the Formula for Exactly One Event:**
- The probability of exactly one of the events \( A \) or \( B \) occurring can be expressed as:
\[ P(A \text{ only}) + P(B \text{ only}) = P(A) \cdot (1 - P(B)) + P(B) \cdot (1 - P(A)) \]
- This can be simplified to:
\[ P(A)(1 - P(B)) + P(B)(1 - P(A)) \]

3. **Substituting the Probabilities:**
- Substitute \( P(A) = p \) and \( P(B) = 2p \):
\[ p(1 - 2p) + 2p(1 - p) \]

4. **Expanding the Expression:**
- Expanding the expression gives:
\[ p - 2p^2 + 2p - 2p^2 = 3p - 4p^2 \]

5. **Setting Up the Equation:**
- We know that this probability equals \( \frac{5}{9} \):
\[ 3p - 4p^2 = \frac{5}{9} \]

6. **Clearing the Fraction:**
- Multiply through by 9 to eliminate the fraction:
\[ 27p - 36p^2 = 5 \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ 36p^2 - 27p + 5 = 0 \]

8. **Using the Quadratic Formula:**
- The quadratic formula is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 36 \), \( b = -27 \), and \( c = 5 \).

9. **Calculating the Discriminant:**
- Calculate \( b^2 - 4ac \):
\[ (-27)^2 - 4 \cdot 36 \cdot 5 = 729 - 720 = 9 \]

10. **Finding the Roots:**
- Substitute back into the quadratic formula:
\[ p = \frac{27 \pm \sqrt{9}}{72} = \frac{27 \pm 3}{72} \]
- This gives us two potential solutions:
\[ p = \frac{30}{72} = \frac{5}{12} \quad \text{and} \quad p = \frac{24}{72} = \frac{1}{3} \]

### Final Answer:
The values of \( p \) are \( \frac{5}{12} \) and \( \frac{1}{3} \).