Question

If A and B are any two events such that P(B) = P(A and B), then which of the following is correct

• A. P(B|A) = 1
• B. P(A|B) = 1
• C. P(B|A) = 0
• D. P(A|B) = 0

Hint:

The condition $P(B) = P(A \cap B)$ implies that event B is a subset of event A ($B \subseteq A$). If you visualize this with a Venn diagram, the entire circle for B is inside the circle for A. Therefore, if B occurs, A must also occur, making $P(A|B) = 1$.

Answer 1

The Correct Option is B

Step 1: Problem Statement: We are provided with information regarding the probabilities of event B and the joint probability of events A and B (denoted as $A \cap B$). Our objective is to calculate a specific conditional probability.
Step 2: Relevant Formulas: The definition of conditional probability is given by:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (\text{if } P(B) > 0) \] \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \quad (\text{if } P(A) > 0) \] Step 3: Derivation: We are given that $P(B) = P(A \cap B)$. We will use this to determine $P(A|B)$.
Using the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Substituting the given condition $P(A \cap B) = P(B)$:
\[ P(A|B) = \frac{P(B)}{P(B)} \] Assuming $P(B) > 0$ (as conditional probability is undefined if $P(B) = 0$):
\[ P(A|B) = 1 \] This result indicates that if event B occurs, event A is guaranteed to occur. This implication is consistent with the condition $P(B) = P(A \cap B)$, which suggests that event B is a subset of event A. Therefore, the occurrence of B necessitates the occurrence of A.
Now, let's examine $P(B|A)$:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{P(B)}{P(A)} \] To determine the value of $P(B|A)$, the value of $P(A)$ is required, which is not provided. Consequently, statements (1) and (3) cannot be definitively confirmed.
Step 4: Conclusion: The established correct statement is $P(A|B) = 1$.