Suppose the events $A$ and $B$ are conditionally independent given $C$. Also, suppose the events $A$ and $B$ are conditionally independent given $C^c$. Under this assumption, will the events $A$ and $B$ be independent? Can you help me either prove or disprove it?
2026-04-25 16:57:44.1777136264
Will conditional independence imply independence under this assumption?
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Imagine that I have two coins in my pocket: a fair coin, and a coin that is biased to land heads $\frac23$ of the time. I take a coin from my pocket and flip it twice. Let:
Then $A,B$ are conditionally independent both given $C$, and given $C^c$: these are two separate flips of the same coin.
But if you use the law of total probability to compute $\Pr[A]$, $\Pr[B]$, and $\Pr[A \cap B]$, you will find that $A$ and $B$ are not independent. The intuition is that the result of one coinflip gives you some hint as to whether the coin is fair, so you gain some information about the other coinflip.