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: Understanding the Concept:
We are given a condition about the probabilities of two events, B and the intersection of A and B (A and B is the same as A $\cap$ B). We need to determine the value of a conditional probability.
Step 2: Key Formula or Approach:
The definition of conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (\text{assuming } P(B)>0) \] \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \quad (\text{assuming } P(A)>0) \] Step 3: Detailed Explanation:
We are given the condition P(B) = P(A $\cap$ B).
Let's use this condition to find P(A|B).
From the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Substitute the given condition P(A $\cap$ B) = P(B) into the formula:
\[ P(A|B) = \frac{P(B)}{P(B)} \] Assuming P(B)>0 (if P(B)=0, the conditional probability P(A|B) is undefined), we get:
\[ P(A|B) = 1 \] This means that given event B has occurred, event A is certain to occur. This makes sense, as the condition P(B) = P(A $\cap$ B) implies that the event B is a subset of the event A. Whenever B happens, A must also happen.
Let's check P(B|A): \[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{P(B)}{P(A)} \] We cannot determine the value of this without knowing P(A). So, option (1) and (3) are not necessarily correct.
Step 4: Final Answer:
The correct statement is P(A|B) = 1.