In probability theory, two events P and Q are said to be independent iff the
probability of their conjunction is equal to the product of their
probabilities. Formally: p(P&Q)=p(P).p(Q).
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.