Two events are independent if knowing the outcome of one does not inform us about the outcome of the other. Formally, two events A and B are independent if the probability p(A&B) that A and B occur together is the product of p(A) times p(B). The concept of independence is crucial, for instance, to evaluating a match between a defendant's DNA and that found on a victim.
Assume only 1 out of 1 million men show such a match. If the DNA of all a country's citizens are in a data bank, and one citizen's DNA is randomly selected, then the probability of a match is about 1 in a million. If the defendant, however, has an identical twin, the probability that the twin also shows a match is 1 (except for procedural errors), not 1 in 1 million. Similarly, if the defendant has brothers, the probability that they match is considerably higher than for the general population. The DNA of relatives is not independent, knowing that one matches increases the chances that the relative also matches.