If I want to find $P(A \cap B)$, is it $1-P(A^\complement \cap B^\complement)$ or $1-P(A \cap B)^\complement$?
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The complement of $A\cap B$ is $(A\cap B)^c$, so $P(A\cap B)=1-P\big((A\cap B)^c\big)$. Using one of the de Morgan’s laws you can go further: $(A\cap B)^c=A^c\cup B^c$.