Explanation

When two events A and B are not independent, the occurrence of one event affects the probability of the other event occurring. In such cases, the joint probability P(A ∩ B) is calculated using the conditional probability formula.

Key Formula:

The conditional probability of A given B is defined as:

$P(A | B) = \frac{P(A \cap B)}{P(B)}$

Rearranging this formula to solve for the joint probability P(A ∩ B):

$P(A \cap B) = P(A | B) \times P(B)$

Why Other Options Are Incorrect:

• Option A (P(A | B)/P(B)): This is the inverse of the correct formula and would give P(A ∩ B)/[P(B)]², which is incorrect.
• Option C (P(A) * P(B)): This formula only applies when events A and B are independent. For independent events, P(A ∩ B) = P(A) × P(B), but the question specifically states the events are NOT independent.
• Option D (None of the above): This is incorrect because Option B provides the correct formula.

Important Distinction:

• Independent events: P(A ∩ B) = P(A) × P(B)
• Dependent events: P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A)

The question explicitly states "if two events are not independent," so we must use the conditional probability formula for dependent events.