Question

If the occurrence of an event $A$ implies the occurrence of an event $B$, then $P(A^C \cap B^C)$ is equal to

• A. $P(B^C)$
• B. $ P(A^C) \, P(B^C)$
• C. $P(A^C)$
• D. $1 - P(A \cap B)$

Answer 1

Answer: (A)

$ P(A^C \cap B^C) = P((A \cup B)^C)$
$ = 1 - (P(A \cup B)$
Since probability of occurrence of an event $A$
implies the occurrence of event $B$
$ \therefore \:\:\:\: A \subset B \Rightarrow A \cup B = B$
$ \Rightarrow \:\: P(A^C \cap B^C ) = 1 - P(B) = P(B^C)$