of the right-hand inequality by P(A), we get P(A)P(A U B) ≤ P(A). Dividing both sides of this by P(A U B) yields P(A) ≤ P(A) / P(A U B) = P(A ∩ (A U B)) / P(A U B) = P(A | A U B), i.e., P(A) ≤ P(A | A U B). When P(A U B) < 1 (i.e., when A U B is a set of less than full probability), the inequality is strict: P(A) < P(A | A U B), and hence, A and A U B are dependent. Note only two assumptions were used in the argument above: (i) 0 < P(A) ≤ 1 which is sufficient to imply P(A) ≤ P(A | A U B). And (ii) P(A U B) < 1, which with (i) implies the strict inequality P(A) < P(A | A U B), and so dependence of A and A U B. It's not necessary to assume A and B are independent--it's true for any events A and B satisfying (i) and (ii) (including independent events). Explanation The cause is that the conditional probability of event occurring, given that it or occurs, is inflated: it is higher than the unconditional probability, because we have excluded cases where neither occur. conditional probability inflated relative to unconditional One can see this in tabular form as follows: the yellow regions are the outcomes where at least one event occurs (and ~A means "not A"). For instance, if one has a sample of , and both and occur independently half the time ( ), one obtains: So in outcomes, either or occurs, of which have occurring. By comparing the conditional probability of to the unconditional probability of : We see that the probability of is higher () in the subset of outcomes where ( or ) occurs, than in the overall population (). On the other hand, the probability of given both and ( or ) is simply the unconditional probability of , , since is independent of . In the numerical example, we have conditioned on being in the top row: Here the probability of is . Berkson's paradox arises because the conditional probability of given within the three-cell subset equals the conditional probability in the overall population, but the unconditional probability within the subset is inflated relative to the unconditional probability in the overall population, hence, within the subset, the presence of decreases