Is the following statement true ?

"If two events E and F are independent, then they are conditionally independent given G ".

I just learned about conditional independence and need to prove or disprove it. Can someone tell me if the above statement is correct. Also can you prove or disprove it formally ?

  • $\begingroup$ It's false. Suppose you are tossing a fair die. Let $E$ be the event "the die comes up even". Let $F$ be the event "the die comes up $>2$". Then let $G$ be the event "the die comes up $>1$". $\endgroup$ – lulu Apr 11 '16 at 14:26

When you conditional on something, actually you are having another probability measure, and independence is a property that depends on which probability measure you are referring to, so in general the statement is incorrect.

A simple example: Consider a coin-flipping experiment, where there are two independent coin flips. Let $E, F$ be the event that head appear in the first and second flip respectively, and $G$ be the event that the total number heads equals to one.


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