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Let $X,Y$ be independent random variables, each with a Bernoulli distribution of parameter $1/2$, and put $S=X+Y$. Unconditionally, the four ordered pairs $(0,0),(0,1),(1,0),(1,1)$ each have probability $1/4$. On the event $S=1$, only $(0,1)$ and $(1,0)$ remain, each with conditional probability $1/2$. ThusKnowing that exactly one trial succeeded makes observing one outcome informative about the other. This violates conditional independence given $S$, even though the original random variables are independent. Given $S=0$ or $S=2$, both variables are constant random variables and are conditionally independent there; that does not repair the failure at $S=1$, an event of positive probability.

The conditional covariance here is $\mathbb{E}[XY\mid S=1]-\mathbb{E}[X\mid S=1]\mathbb{E}[Y\mid S=1]=0-1/4=-1/4$. Both conditional variances are $1/4$, so the conditional correlation coefficient is $-1$: within the selected group, $Y=1-X$. These conclusions follow from restricting a finite sample space, without positing any physical interaction between the trials.

This example and the shared-environment example establish both logical directions: independent random variables need not be conditionally independent, and conditionally independent variables need not be independent. When $S$ is represented as the joint result of two inputs in a causal directed acyclic graph, conditioning on it has the collider structure associated with collider bias. A probability calculation by itself establishes a dependence relation, not a physical causal mechanism or an intervention.

A constant random variable takes the same fixed real number $c$ at every outcome in its sample space. Its probability distribution puts mass one at $c$. An almost surely constant random variable may differ on an event of probability zero, while having the same probability distribution.