Question

Given that A and B are not null sets, which of the following statements regarding probability is/are CORRECT?

• A. \( P(A \cap B) = P(A) \cdot P(B) \), if A and B are mutually exclusive.
• B. Conditional probability, \( P(A \mid B) = 1 \) if \( B \subseteq A \).
• C. \( P(A \cup B) = P(A) + P(B) \), if A and B are mutually exclusive.
• D. \( P(A \cap B) = 0 \), if A and B are independent.

Hint:

Mutual exclusivity implies \( P(A \cap B) = 0 \), while independence implies \( P(A \cap B) = P(A) \cdot P(B) \). These two are not the same and should not be confused.

Answer 1

The Correct Option is B, C

Option (A):
If A and B are mutually exclusive, then: \[ P(A \cap B) = 0 \] So the expression \( P(A \cap B) = P(A) \cdot P(B) \) is incorrect unless A and B are independent, not mutually exclusive. Hence, option (A) is incorrect.
Option (B):
Conditional probability is defined as: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] If \( B \subseteq A \), then \( A \cap B = B \), so: \[ P(A \mid B) = \frac{P(B)}{P(B)} = 1 \] Thus, option (B) is correct.
Option (C):
If A and B are mutually exclusive, then: \[ P(A \cap B) = 0 \Rightarrow P(A \cup B) = P(A) + P(B) \] So, option (C) is correct.
Option (D):
If A and B are independent, then: \[ P(A \cap B) = P(A) \cdot P(B) \] This value is generally non-zero unless either \( P(A) \) or \( P(B) \) is zero. Hence, option (D) is incorrect.