If I want to find $P(A \cap B)$, is it $1-P(A^\complement \cap B^\complement)$ or $1-P(A \cap B)^\complement$?
Probability - Finding opposite of complement
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0A Venn diagram will let you quickly identify $A^c\cap B^c$ and $(A\cap B)^c$ visually. Then you can *see* that $1-P((A\cap B)^c)$ is right, and also *see* why the other is in general not right. – 2012-10-13
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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: $(A\cap B)^c=A^c\cup B^c$.