Random variables X and Y are dependent conditioned on random variable Z I intuitively understand the concept of "X and Y are independent conditioned on Z", but I don't get the concept of "X and Y are dependent conditioned on Z". Can you provide some examples which show this case?
 A: Suppose we have the following structural casual model

which is the SCM of the famous Monty Hall problem, where


*

*$X$ is the door chosen by the player 

*$Y$ is the door hiding the car

*$Z$ is the door chosen by Monty (the host of the program).


In this problem, there are three doors: $A, B$ and $C$. Two of these doors hide a goat and the other hides a Ferrari. The goal of the game is, of course, to choose the door with the Ferrari. Initially, the player choses a door, $X$, (among three doors $A, B$ and $C$), without opening it, so it doesn't know if $X$ hides a goat or a Ferrari. Then, the host, called "Monty", opens another door (which is different from $X$), $Z$, and it shows to the player that the door contains a goat. The problem is: should the player switch door or not? (The answer is "yes", but it doesn't matter much to understand the concept asked in the question).
In the SCM above, $U_X, U_Y$ and $U_Z$ are the variables which are external (exogenous), whose causes are not explained. In the model above, a direct edge between $X$ and $Z$ means that $X$ is a cause of $Z$. $Z$ also depends on $X$, that is, the door chosen by Monty depends on the door chosen by the player: Monty will choose the door that does not contain the Ferrari. Note that Monty of course knows what is behind each door.
In the SCM above, $X$ and $Y$ are dependent conditionally on $Z$, because, if we know the value of $Z$ (i.e. the door chosen by Monty) and the value of $X$ (the door chosen by the player), then the door containing the Ferrari depends on (or is determined by) the door chosen by the player, i.e. $X$.
In a certain way, and maybe intuitively, we are restricting the possibilities by conditioning on $Z$.
It is worth noting that $X$ and $Y$ are dependent conditioned on $Z$, but they are not causes of each other (as the SCM shows).
See also: https://en.wikipedia.org/wiki/Collider_(epidemiology).
A: Let $Z$ be gender and half male and half female, $X$ be smoking and half smoking and half non-smoking, $Y$ be cancer.
For male, we have 
                          Cancer
                     Yes           No       total   
       Smoking        0.3         0.2       0.5
       Non-smoking    0.2         0.3       0.5
       Total          0.5         0.5       1.0

For female, we have 
                          Cancer
                     Yes           No       total   
       Smoking        0.2         0.3       0.5
       Non-smoking    0.3         0.2       0.5
       Total          0.5         0.5       1.0

Put male and female together, we have
                          Cancer
                     Yes           No       total   
       Smoking        0.25         0.25       0.5
       Non-smoking    0.25         0.25       0.5
       Total          0.50         0.50       1.0

So "X and Y are dependent conditioned on Z", and X and Y are independent un-conditioned on Z.
A: Here's another example, which is explained here and was apparently first given by Pearl in 1988.
Suppose there are two independent causes of "your car refusing to start" ($Z$): "having a dead battery" ($X$) and "having no gas" ($Y$).
We can model this as a casual model:

You can ignore the random variables $U_X, U_Y$ and $U_Z$, which are the exogenous variables (i.e. the "errors"). This casual model is what is called a "collider".
Given that the causes of $Z$, i.e. $X$ ("having no battery") and $Y$ ("having no gas"), are independent of each other, telling you that the car has no battery will not tell you anything about the car having or not gas. However, if I tell you that the car won't start and that the battery is not dead, then the car doesn't start because the gas tank must be empty. In other words, two independent causes can be made dependent by conditioning on the "effect" (i.e. a collider) of the two causes. So, if we know the effect and one of the causes, then we also know the other cause.
To clarify, in practice, "conditioning on a random variable $Z$" means to give information about the "realisation" of $Z$.
A: We can answer this question also using a slightly different approach. We can think of the "conditioning" operation as a "filtering" operation in a dataset. 
Suppose we have the usual collider $X \rightarrow Z \leftarrow Y$ (which is apparently if not the only the most common situation where this occurs). When we condition on $Z=z$ (i.e. when we fix the value of the variable $Z$ to be $z$), we limit our comparisons (in the dataset) to cases in which Z takes the value $z$. But Z depends, for its value, on $X$ and $Y$. Therefore, if we change $X$, given that $X$ is a cause of $Z$ and given that $Z$ is now a constant (i.e. cannot be changed), then, to compensate this change in $X$, $Y$ must also change. Note that, in general, if a cause changes, the effect should also change.
For example, suppose that $Z = X + Y$ and that $X$ and $Y$ are independent, that is, the value of $X$ does not depend on the value of $Y$ (and vice-versa). If $X=3$, in general, we cannot learn anything about $Y$ (because they are independent). However, if I fix $Z=10$ (i.e. we cannot change $Z$ anymore), then if I change $X$ (e.g. $X=4$) I must also change $Y$, in order to maintain $Z=10$.
