If the occurrence of an event A implies the occurrence of an event B, then P(AC ∩ BC) is equal to

Question

If the occurrence of an event A implies the occurrence of an event B, then P(AC ∩ BC) is equal to (A) P(BC) (B)  P(AC) P(BC) (C) P(AC) (D) 1 - P(A ∩ B)

Tardigrade Admin · Accepted Answer

P(AC ∩ BC) = P((A ∪ B)C) = 1 - (P(A ∪ B) Since probability of occurrence of an event A implies the occurrence of event B ∴ : :: : A ⊂ B ⇒ A ∪ B = B ⇒ : : P(AC ∩ BC ) = 1 - P(B) = P(BC)

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

$P (B^{C})$

$P (A^{C}) P (B^{C})$

$P (A^{C})$

$1 - P (A \cap B)$

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

Q. If the occurrence of an event A implies the occurrence of an event B, then P(AC∩BC) is equal to

P(BC)

P(AC)P(BC)

P(AC)

1−P(A∩B)