Probability - the Australian Mathematical Sciences Institute

{26} • ProbabilityIndependenceThe concept of independence of events plays an important role throughout probabilityand statistics. The most helpful way to think about independent events is as follows.We first consider two events A and B, and assume that Pr(A) ≠ 0 and Pr(B) ≠ 0. Theevents A and B are independent ifPr(A|B) = Pr(A).This equation says that the conditional probability of A, given B, is the same as the unconditionalprobability of A. In other words, given that we know that B has occurred, theprobability of A is unaffected.Using the rule for conditional probability, we see that the events A and B are independentif and only ifPr(A ∩ B) = Pr(A) × Pr(B).This equation gives us a useful alternative characterisation of independence in the caseof two events. The symmetry of this equation shows that independence is not a directionalrelationship: the events A and B are independent if Pr(B|A) = Pr(B). So, for independentevents A and B, whether B occurs has no effect on the probability that A occurs,and similarly A has no effect on the probability that B occurs.If events A and B are not independent, we say that they are dependent. This does notnecessarily mean that they are directly causally related; it just means that they are notindependent in the formal sense defined here.Events that are physically independent are also independent in the mathematical sense.Mind you, physical independence is quite a subtle matter (keeping in mind such phenomenaas the ‘butterfly effect’ in chaos theory, the notion that a butterfly flapping itswings in Brazil could lead to a tornado in Texas). But we often take physical independenceas the working model for events separated by sufficient time and/or space. Wemay also assume that events are independent based on our consideration of the independenceof the random processes involved in the events. Very commonly, we assumethat observations made on different individuals in a study are independent, because theindividuals themselves are separate and unrelated.For example, if A = “my train is late” and B = “the clue for 1 across in today’s cryptic crosswordinvolves an anagram”, we would usually regard A and B as independent. However,if C = “it is raining this morning”, then it may well be that A and C are not independent,while B and C are independent.While physical independence implies mathematical independence, the converse is nottrue. Events that are part of the same random procedure and not obviously physically