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So I'm studying Probability and I'm having a hard time figuring out how to find the cardinality of unions and intersections given certain information. Specifically, given a sample space $ \Omega $ and three events $A,B,C$ :

\begin{gathered} n(\Omega ) = 270 \hfill \\ n(A) = n(B) = n(C) = 90 \hfill \\ n(A \cap B) = n(A \cap C) = n(B \cap C) = 30 \hfill \\ n(A \cap B \cap C) = 10 \hfill \\ \end{gathered}

How do I find the cardinality of the intersection of the three complements? $ n({A^c} \cap {B^c} \cap {C^c}) \\ $

I could solve it very easily drawing a Venn diagram but I want to know how to solve it analytically.

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$$n(A^c\cap B^c\cap C^c) = n((A\cup B\cup C)^c)=n(\Omega)-n(A\cup B\cup C)$$

Use PIE to calculate $n(A\cup B\cup C)$

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  • $\begingroup$ thanks! this literally came into my mind a couple minutes ago; I will be posting the answer $\endgroup$ – Maximiliano Santiago Jun 24 '17 at 6:01
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\begin{gathered} n({A^c} \cap {B^c} \cap {C^c}) = n({(A \cup B \cup C)^C}) = 270 - n(A \cup B \cup C) \hfill \\ {\text{See that }}n(A \cup B \cup C) = 270 - 90 + 10 = 190 \hfill \\ {\text{Thus, }}n({A^c} \cap {B^c} \cap {C^c}) = n({(A \cup B \cup C)^C}) = 270 - 190 = 80 \hfill \\ \end{gathered}

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