# Can a transcript change dependent random variables into independent variables?

Let's say $$X, Y$$ are dependent random variables. Can I find an example such that for a transcript $$t$$ conditioned on a communication protocol $$\Pi=t$$, the variables become independent?

Yes. Consider the following protocol. Alice flips a coin, and sets $$X$$ and $$Y$$ to be equal to the outcome of the coin flip. She then sends $$X,Y$$ to Bob. Note that the transcript will include the value of $$X,Y$$.
Now unconditionally, $$X,Y$$ are dependent (there is probability $$1/2$$ that they are Heads,Heads and probability $$1/2$$ that they are Tails,Tails). But conditioned on the transcript, $$X,Y$$ are independent (e.g., if the transcript says they are Heads, then with conditional probability $$1$$ they are Heads,Heads and all other cases have conditional probability $$0$$; that meets the definition for independence).