Question

If \( P(B) \neq 0 \) and \( P(A | B) = 1 \) for two events \( A \) and \( B \), then ____________.

Hint:

When \( P(A | B) = 1 \), it implies that event \( B \) is entirely contained within event \( A \), or \( B \subseteq A \).

Answer 1

Step 1: We are given that \( P(B) \neq 0 \) and \( P(A | B) = 1 \). The conditional probability \( P(A | B) \) is defined as: \[ P(A | B) = \frac{P(A \cap B)}{P(B)}. \] Since \( P(A | B) = 1 \), this implies: \[ \frac{P(A \cap B)}{P(B)} = 1. \] Multiplying both sides by \( P(B) \), we get: \[ P(A \cap B) = P(B). \] Step 2: The equation \( P(A \cap B) = P(B) \) means that the probability of the intersection of \( A \) and \( B \) is equal to the probability of \( B \). This implies that every outcome in \( B \) is also in \( A \), i.e., \( B \subseteq A \). Thus, the correct conclusion is: \[ B \subseteq A. \]