Question

If A and B are two events, then \(P(A|B)\) is:

• A. \(\frac{P(A \cap B)}{P(B)} \text{ if } P(B) > 0\)
• B. \(\frac{P(A \cap B)}{P(B)} \text{ if } P(A) > 0\)
• C. \(\frac{P(A \cap B)}{P(A)} \text{ if } P(A) > 0\)
• D. \(\frac{P(A \cap B)}{P(A)} \text{ if } P(B) > 0\)

Hint:

Exam Tip:
Conditional probability formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] The denominator must be positive.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Conditional probability \(P(A|B)\) is the probability of event \(A\) occurring given that event \(B\) has already occurred. 

Step 2: Key Formula or Approach:
The formula for conditional probability is: \[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{provided } P(B) > 0 \] The condition \(P(B) > 0\) is necessary because we cannot condition on an event with zero probability. 

Step 3: Analyzing the Options:
(A) \(\frac{P(A \cap B)}{P(B)}\), if \(P(B) > 0\): Correct. 
(B) \(\frac{P(A \cap B)}{P(B)}\), if \(P(A) > 0\): Incorrect condition. 
(C) \(\frac{P(A \cap B)}{P(A)}\), if \(P(A) > 0\): This represents \(P(B|A)\), not \(P(A|B)\). 
(D) \(\frac{P(A \cap B)}{P(A)}\), if \(P(B) > 0\): Incorrect formula. 

Step 4: Final Answer:
Therefore, option (A) is correct.