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(C) Two events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A) \cdot P(B)$.By De Morgan's Law,$A' \cap B' = (A \cup B)'$.Therefore,$P(A

Vedclass · Accepted Answer

$P(A' \cap B') = [1 - P(A)][1 - P(B)]$

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(C) Two events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A) \cdot P(B)$.By De Morgan's Law,$A' \cap B' = (A \cup B)'$.Therefore,$P(A' \cap B') = 1 - P(A \cup B)$.Using the addition theorem of probability,$P(A \cup B) = P(A) + P(B) - P(A \cap B)$.Since $A$ and $B$ are independent,$P(A \cup B) = P(A) + P(B) - P(A)P(B)$.Substituting this into the expression for $P(A' \cap B')$:$P(A' \cap B') = 1 - [P(A) + P(B) - P(A)P(B)]$$P(A' \cap B') = 1 - P(A) - P(B) + P(A)P(B)$$P(A' \cap B') = [1 - P(A)] - P(B)[1 - P(A)]$$P(A' \cap B') = [1 - P(A)][1 - P(B)]$.Thus,option $C$ is correct.

For two independent events $A$ and $B$,which of the following is true?

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