TITLE: How to prove independence is not a transitive relation?
QUESTION [5 upvotes]: Suppose $A$ and $B$ are independent, and $B$ and $C$ are independent.
Is $B$ independent of $A \cap C$?  Is $B$ independent of $A \cup C$? If so, prove. If not so, give a counterexample.
Could someone give some hints on how to prove them? I am quite confused about them.

REPLY [5 votes]: I flip two coins. Let $A$ be the event that the first coin lands heads up, and let $B$ be the event the second coin lands heads up. Let $C$ be the same event as $A$!
$A$ and $B$ are independent, and $B$ and $C$ are independent, but $A$ and $C$ are as far from independent as it gets.