Philosophers such as Hugh Mellor[1] and Patrick Suppes[2] have defined causation in terms of a cause preceding and increasing the probability of the effect.
In general, formulating the notion of "probability raising" within the calculus of do-operators[4] resolves the difficulties that probabilistic causation has encountered in the past half-century,[2][5][6] among them the infamous Simpson's paradox, and clarifies precisely what relationships exist between probabilities and causation.
The establishing of cause and effect, even with this relaxed reading, is notoriously difficult, expressed by the widely accepted statement "Correlation does not imply causation".
In statistics, it is generally accepted that observational studies (like counting cancer cases among smokers and among non-smokers and then comparing the two) can give hints, but can never establish cause and effect.
(This is a common criticism of studies of safety of food additives that use doses much higher than people consuming the product would actually ingest.)
In a closed system the data may suggest that cause A * B precedes effect C in a defined interval of time τ.
One can reasonably claim, within 6 Standard Deviations, that A * B cause C given the time boundary (such as 50 ms, or 50 hours) IF And Only IF A, B and C are the only parts of the system in question.