Question

If P(A | B) < P(A), then P(B | A) is

• A. < P(B)
• B. > P(B)
• C. = P(B)
• D. $\neq$ P(B)

Hint:

If learning that $B$ has occurred decreases the probability of $A$, then learning that $A$ has occurred must also decrease the probability of $B$. This symmetry holds true for all probability relationships.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Conditional probability defines the likelihood of an event occurring given that another event has already occurred.
Key Formula or Approach:
The definition of conditional probability is:
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \implies P(A \cap B) = P(A \mid B) \cdot P(B) \]
Similarly:
\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \implies P(A \cap B) = P(B \mid A) \cdot P(A) \]

Step 2: Detailed Explanation:
We are given the inequality:
\[ P(A \mid B) < P(A) \]
Substitute the definition $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$:
\[ \frac{P(A \cap B)}{P(B)} < P(A) \]
Multiply both sides of the inequality by $P(B)$ (since $P(B) \ge 0$):
\[ P(A \cap B) < P(A) \cdot P(B) \]
Now, substitute $P(A \cap B) = P(B \mid A) \cdot P(A)$ into this inequality:
\[ P(B \mid A) \cdot P(A) < P(A) \cdot P(B) \]
Divide both sides by $P(A)$ (assuming $P(A) > 0$):
\[ P(B \mid A) < P(B) \]
Therefore, $P(B \mid A)$ is less than $P(B)$.

Step 3: Final Answer
The correct option is (A).