Consider three real random variables $X$, $Y$, and $Z$. Suppose
- $X$ is independent with $Y$, and
- $X$ is not independent with $Z$.
Does this imply that $Y$ and $Z$ are independent?
The above seems intuitive: Since $X$ and $Z$ are not independent, $Z$ has information on $X$. If $Y$ and $Z$ are not independent, then $Y$ has information on $Z$, and thus has information on $X$, which contradicts the fact that $X$ and $Y$ are independent.
No, that's wrong. Consider $X,Y$ the results of fair dice rolls, and take $Z=X+Y$. The idea here is that $Y$ gives some partial information on $Z$, but which is insufficient to derive anything about it -- $X$ and $Y$ give "different" kinds of information to $Z$.