Question

If \( A \) and \( B \) are independent events and \( P\left((A - B) \cup (B - A)\right) = k - P(A)P(B) \), then \( k = \) 

• A. \(\text{P}(A) - \text{P}(B)\)
• B. \(\frac{\text{P}(A)}{\text{P}(B)}\)
• C. \(\text{P}(A)\text{P}\left(\frac{B}{A}\right)\)
• D. \(\text{P}(A) + \text{P}(B)\)

Hint:

Conditional probability of independent events is unaffected by the condition.
Always translate slash notations like \(\frac{A}{B}\) to standard notation \(A|B\) to avoid confusion with division.

Answer 1

The Correct Option is D

Concept:
• For independent events, conditional probability simplifies as: \(\text{P}(A|B) = \text{P}(A)\) and \(\text{P}(B|A) = \text{P}(B)\).
• Note that notation \(\frac{A}{B}\) represents the conditional probability \(A|B\).
• The addition rule for any two probabilities is: \(\text{P}(X \cup Y) = \text{P}(X) + \text{P}(Y) - \text{P}(X \cap Y)\).
• For independent conditional probabilities, \(\text{P}((A|B) \cap (B|A)) = \text{P}(A) \cdot \text{P}(B)\).

Step 1: Simplify the conditional components using independence
Given \(A\) and \(B\) are independent: \[ \text{P}\left(\frac{A}{B}\right) = \text{P}(A|B) = \text{P}(A) \] \[ \text{P}\left(\frac{B}{A}\right) = \text{P}(B|A) = \text{P}(B) \]

Step 2: Apply the union formula to the given expression
\[ \text{P}\left(\left(\frac{A}{B}\right) \cup \left(\frac{B}{A}\right)\right) = \text{P}(A) + \text{P}(B) - \text{P}\left(\left(\frac{A}{B}\right) \cap \left(\frac{B}{A}\right)\right) \] Since the events occur independently: \[ = \text{P}(A) + \text{P}(B) - \text{P}(A)\text{P}(B) \]

Step 3: Compare with the given expression to solve for \(k\)
The problem statement gives: \[ \text{P}\left(\left(\frac{A}{B}\right) \cup \left(\frac{B}{A}\right)\right) = k - \text{P}(A)\text{P}(B) \] Comparing the two expressions: \[ k = \text{P}(A) + \text{P}(B) \]