Question

Let $E$ and $F$ be two events. Then which one of the following statements is NOT always TRUE?

• A. $P(E \cap F) \le \max\{1 - P(E^C) - P(F^C), 0\}$
• B. $P(E \cup F) \ge \max\{P(E), P(F)\}$
• C. $P(E \cup F) \le \min\{P(E) + P(F), 1\}$
• D. $P(E \cap F) \le \min\{P(E), P(F)\}$

Hint:

In probability, the range of $P(E \cap F)$ is always between 0 and $\min(P(E), P(F))$, while $P(E \cup F)$ ranges between $\max(P(E), P(F))$ and $\min(P(E) + P(F), 1)$.

Answer 1

The Correct Option is A

Step 1: Recall probability laws.
We know: \[ P(E \cup F) = P(E) + P(F) - P(E \cap F). \] Also, \[ 0 \le P(E \cap F) \le \min(P(E), P(F)). \]

Step 2: Checking each option.
(B) is true because union probability is always greater than or equal to the maximum of individual probabilities.
(C) is true since $P(E \cup F)$ cannot exceed $1$ or $P(E) + P(F)$.
(D) is true since intersection cannot exceed either individual probability.
(A) is not always true because the inequality involving complements does not hold for all probability combinations — it can fail for certain $E$, $F$ with small intersection or independence.

Step 3: Conclusion.
Hence, statement (A) is NOT always true.