Conditional independence in three events

Question

Let X, Y, Z be three events in a probability space. Suppose that X and Y are conditionally independent given Z. Can we say that X and Y are conditionally independent given Z'(Z complement)? Can anyone give a counter example or valid proof?

Answer 1

Suppose you have the following objects of different shapes, sizes, and colors:

$1$ Large Blue Square

$1$ Large Blue Circle

$1$ Large Red Square

$1$ Large Red Circle

$1$ Small Blue Square

$2$ Small Blue Circles

$1$ Small Red Square

$1$ Small Red Circle

Suppose you randomly p[ick one of these objects, and define the following events:

$X$: the object is Blue

$Y$: the object is a Square

$Z$: the object is Large

It is easy to see that $X$ and $Y$ are independent given $Z$: if the object picked is Large, then the probability of picking a Blue object is $50%$, no matter whether the given objects is a Square or a Circle.

However, $X$ and $Y$ are not independent given $Z'$: if the object you picked is not Large (i.e. Small), then the probability of picking a Blue object is higher if you also know you have picked a Circle, as opposed to having picked a Square.