In probability calculus there is no expression for causal dependency. No one can express with its semantics the popular example that manipulation on barometer do not change the weather, but changes of the weather change barometer measurements. Either two events 'tend to occurs together' (correlate) or not.
The very definition of independence is (probably) derived from the idea, that knowledge if $B$ occurred, do not change probability of occurring event $A$. This is formally written as $P(A) = P(A|B)$.
The contradiction to situation state is lack of independency: the probability of occurring event $B$ increases or decreases probability of occurring event $A$. This is true for barometer and the weather and expressed as $P(A) \neq P(A|B)$.
Mathematicians often know, that their not independency is not always the 'true' dependency and restrain themselves from using causally marked expression. Especially, that in econometrics or causal inference such definition is exist. Therefore at some probability calculus courses you would hear, that no one discussed $dependency$, the discussed ideas were not independency and correlation.
The mathematical tool which analyses dependency in the more natural meaning is do-calculus (by Judea Pearl). This tool extends standard probability calculus with the do operator, which describes intervention in the system. For the barometer and the weather all four statements will be true:
$$P(A) \neq P(A|B)$$
$$P(B) \neq P(B|A)$$
$$P(B) \neq P(B|do(A))$$
$$P(A) = P(A|do(B))$$
In this context I would strongly discourage using word dependent in context of standard probability calculus and statistics. Not independent is good enough, and in fact more precise in context of this 'more advanced' mathematics.