I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ Intuitively, it looks true but why is that?
2026-03-31 19:16:46.1774984606
Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$
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This is not true. Say $\Omega=\{(0,0),(0,1),(1,0),(1,1)\}$. Say each point of $\Omega$ has probability $1/4$. Let $A=\{(0,0),(1,0)\}$, $B=\{(0,0),(0,1)\}$, and $C=\{(0,0),(1,1)\}$.