Question

For two events $A$ and $B$, a true statement among the following is

• A. $P(\overline{A} \cup \overline{B}) = 1 - P\left(\dfrac{B}{A}\right)$
• B. $P(\overline{A} \cup \overline{B}) = 1 - P(A \cup B)$
• C. $P(\overline{A} \cup \overline{B}) = P(A \cup B)$
• D. $P(\overline{A} \cup \overline{B}) = P(A) + P(\overline{B})$

Hint:

Review relationships among union, intersection, and complement of events for probability identities.

Answer 1

The Correct Option is A

Using set complement identity:
$\overline{A} \cup \overline{B} = \overline{A \cap B}$
Therefore, $P(\overline{A} \cup \overline{B}) = 1 - P(A \cap B)$
If $\dfrac{B}{A}$ represents conditional probability, then
$P(B|A) = \dfrac{P(A \cap B)}{P(A)} \Rightarrow P(A \cap B) = P(A) \cdot P(B|A)$
This matches structure of answer (1) if interpreted properly