We know from statistics that the notion of statistical independence says that the joint probability of two random variables A and B is just the product of their (marginal) probabilities. Sometimes, two variables may not be statistically independent of each other to begin with, but observing a third variable, C, might result in them becoming independent of each other. In short, we say that events A and B are conditionally independent given C, and we can express this as:

For example, suppose that J represents the probability of being given a job offer at a particular company and G represents the probability of being accepted into graduate school at a particular university. Both of these might depend on a variable U, a person's performance on their undergraduate degree. This can be summarized in a graph as:

When we don't know U, a person's performance on their undergraduate degree, knowing that they were accepted into graduate school might increase our belief in their chances of getting a job and vice versa. This is because we are inclined to believe that they did well in their undergraduate degree, which influences that person's chances of getting a job. Thus, the two events J and G are not independent of each other.

If we are told the performance of a person on their undergraduate degree, however, we might assume that the person's chance of getting a job offer might be independent of their chance of getting into graduate school. This is because of other factors that might affect this, such as the person's job interview on a particular day or the quality of other potential candidates for the job, which are not influenced by the person's application to graduate school.