Since in probability theory p(Q|P)=p(P&Q):pr(P) i.e. the probability of Q given P equals the probability of P&Q divided by the probability of P, it follows that if 1>p(P)>0, then P is - probabilistically - independent of Q iff p(Q|P) = p(Q|~P) = p(Q).

In practice and in theory, many things are supposed, both explicitly and implicitly, to be independent that in fact are not, while also often small dependencies, when p(Q) does not differ much from p(Q|P), are neglected.

Many statistical tests and procedures involve somewhere one or more hypotheses of independence, because this tends to much simplify the formulas and reasoning, which may have been explicit hypotheses in statistical theory, but often are disregarded in the practice of statistics, when formulas and statistical tests are applied by rote and recipe, without much knowledge, understanding or care for their theoretical underpinnings and assumptions.