Question

If \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \), prove that the events \( A \) and \( B \) are independent.

Hint:

To prove the independence of two events, verify that \( P(A \cap B) = P(A) . P(B) \).

Answer 1

Step 1: Formula for Probability of Union.
The formula for the probability of the union of two events \( A \) and \( B \) is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]
Step 2: Substitute the Given Values.
We are given that \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A \cup B) = \frac{2}{3} \). Substituting these values into the formula: \[ \frac{2}{3} = \frac{1}{2} + \frac{1}{3} - P(A \cap B). \]
Step 3: Solve for \( P(A \cap B) \).
To solve for \( P(A \cap B) \), first find the sum of \( \frac{1}{2} \) and \( \frac{1}{3} \): \[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}. \] Now substitute into the equation: \[ \frac{2}{3} = \frac{5}{6} - P(A \cap B). \] Solving for \( P(A \cap B) \): \[ P(A \cap B) = \frac{5}{6} - \frac{2}{3} = \frac{5}{6} - \frac{4}{6} = \frac{1}{6}. \]
Step 4: Check for Independence.
Two events \( A \) and \( B \) are independent if: \[ P(A \cap B) = P(A) . P(B). \] We know that \( P(A) = \frac{1}{2} \) and \( P(B) = \frac{1}{3} \). Therefore: \[ P(A) . P(B) = \frac{1}{2} . \frac{1}{3} = \frac{1}{6}. \] Since \( P(A \cap B) = \frac{1}{6} \), we conclude that \( A \) and \( B \) are independent.
Step 5: Conclusion.
Hence, the events \( A \) and \( B \) are independent.