Question

If \( A \) and \( B \) are two events such that \( A \subseteq B \) and \( P(B) \neq 0 \), then which of the following is correct?

• A. \( P(A)<P(B) \)
• B. \( P(A | B) \geq P(A) \)
• C. \( P(A) = P(B) \)
• D. \( P(A | B) = \frac{P(A)}{P(B)} \)

Hint:

When one event is a subset of another, the conditional probability \( P(A | B) \) is always greater than or equal to \( P(A) \), since \( P(A) \leq P(B) \).

Answer 1

The Correct Option is B

Given that \( A \subseteq B \), it implies that the probability of \( A \) occurring is always less than or equal to the probability of \( B \). The conditional probability \( P(A | B) \) is given by: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \] Since \( A \subseteq B \), \( A \cap B = A \), so: \[ P(A | B) = \frac{P(A)}{P(B)} \] Also, since \( P(A) \leq P(B) \), it follows that: \[ P(A | B) \geq P(A) \] Thus, the correct answer is option (2).