- #1

mathmari

Gold Member

MHB

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Let $A,B,C$ be independent events. I want to show that $A^c, B^c, C^c$ are independent.

We have the following: \begin{align*}&P(A^c\cap B^c\cap C^c)=1-P(A\cup B\cup C)\\ & =1-\left [P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)-P(A\cap B\cap C)\right ] \\ & = 1-\left [P(A)+P(B)+P(C)-P(A)P(B)-P(A)P(C)-P(B)P( C)-P(A)P (B)P( C)\right ] \\ & = 1-P(A)-P(B)-P(C)+P(A)P(B)+P(A)P(C)+P(B)P( C)+P(A)P (B)P( C) \\ & = 1+P(A)\left [P(B)-1\right ]+P(C)\left [P(A)-1\right ]+P(B)\left [P( C)-1\right ]+P(A)P (B)P( C) \\ & = 1-P(A)P(B^c)-P(C)P(A^c)-P(B)P( C^c)+P(A)P (B)P( C)\end{align*} Is everything correct so far? How could we continue? (Wondering)