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UIUC STAT 400 - 408general2

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Math 408, Actuarial Statistics I A.J. HildebrandGeneral Probability, II: Independence and conditional proba-bilityDefinitions and properties1. Independence: A and B are called independent if they satisfy the product formulaP (A ∩ B) = P (A)P (B).2. Conditional probability: The conditional probability of A given B is denoted byP (A|B) and defined by the formulaP (A|B) =P (A ∩ B)P (B),provided P (B) > 0. (If P (B) = 0, the conditional probability is not defined.)3. Independence of complements: If A and B are independent, then so are A andB0, A0and B, and A0and B0.4. Connection between independence and conditional probability: If the con-ditional probability P (A|B) is equal to the ordinary (“unconditional”) probabilityP (A), then A and B are independent. Conversely, if A and B are independent, thenP (A|B) = P (A) (assuming P (B) > 0).5. Complement rule for conditional probabilities: P (A0|B) = 1 − P (A|B). Thatis, with respect to the first argument, A, the conditional probability P (A|B) satisfiesthe ordinary complement rule.6. Multiplication rule: P (A ∩ B) = P (A|B)P (B)Some special cases• If P (A) = 0 or P (B) = 0 then A and B are independent. The same holds whenP (A) = 1 or P (B) = 1.• If B = A or B = A0, A and B are not independent except in the above trivial casewhen P (A) or P (B) is 0 or 1. In other words, an event A which has probabilitystrictly between 0 and 1 is not independent of itself or of its complement.Notes and hints• Formal versus intuitive notion of independence: When working problems,always use the above formal mathematical definitions of independence and conditionalprobabilities. While these definitions are motivated by our intuitive notion of theseconcepts and most of the time consistent with what our intuition would predict,intuition, aside from being non-precise, does fail us some time and lead to wrongconclusions as illustrated, for example, by the various paradoxes in probability.1Math 408, Actuarial Statistics I A.J. Hildebrand• Independence is not the same as disjointness: If A and B are disjoint (corre-sponding to mutually exclusive events), then the intersection A ∩ B is the empty set,so P (A∩B) = P (∅) = 0, so independence can only hold in the trivial case when one ofthe events has probability 0. While, at first glance, this might seem counterintuitive,it is, in fact, consistent with the interpretation of disjointness as meaning that A andB are mutually exclusive, that is, if A occurs, then B cannot occur, and vice versa.• Independence and conditional probabilities in Venn diagrams: In contrastto other prop erties such as disjointness, independence can not be spotted in Venndiagrams. On the other hand, conditional probabilities have a natural interpretationin Venn diagrams: The conditional probability given B is the probability you get ifthe underlying sample space S is “shrunk” to the set B (i.e., everything outside B isdeleted), and then rescaled so as to have again unit area.• Verbal descriptions of conditional probabilities: Whether a probability in aword problem represents a conditional or ordinary (unconditional) probability is notalways obvious, and you have to read the problem carefully to see which interpretationis the correct one. Typically, conditional probabilities are indicated by words like“given”, “if”, or “among” (e.g., in the context of subpopulations), though there areno hard rules, and it may depend on what the underlying universe (sample space)is, which is usually not explicitly stated, but should be clear from the context of theentire problem.• Don’t make assumptions about independence: If a problem does not explicitlystate that two events are independent, they are probably not, and not you should notmake any assumptions about independence.• P (A|B) is not the same as P (B|A): In contrast to set-theoretic operations likeunion or intersection, in conditional probabilities the order of the sets matters.• P (A|B0) is not the same as 1 − P (A|B): The complement formula only holds withrespect to the first argument. There is no corresponding formula for P (A|B0).• Independence of three or more events: Though rarely needed, the definition ofindependence of two events can be extended to three events as follows: A, B, C arecalled mutually independent if the product formula holds for (i) the intersectionof all three events (i.e., P (A ∩B ∩C) = P (A)P (B)P (C)) and (ii) for any combinationof two of these three events (i.e., P (A ∩ B) = P (A)P (B) and similarly for P (A ∩ C),P (B ∩ C)). More generally, n events A1, . . . , Anare called independent if the productformula holds for any subcollection of these


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UIUC STAT 400 - 408general2

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