Math 408 Actuarial Statistics I A J Hildebrand General Probability II Independence and conditional probability Definitions and properties 1 Independence A and B are called independent if they satisfy the product formula P A B P A P B 2 Conditional probability The conditional probability of A given B is denoted by P A B and defined by the formula P 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 and B 0 A0 and B and A0 and B 0 4 Connection between independence and conditional probability If the conditional probability P A B is equal to the ordinary unconditional probability P A then A and B are independent Conversely if A and B are independent then P A B P A assuming P B 0 5 Complement rule for conditional probabilities P A0 B 1 P A B That is with respect to the first argument A the conditional probability P A B satisfies the 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 when P A 1 or P B 1 If B A or B A0 A and B are not independent except in the above trivial case when P A or P B is 0 or 1 In other words an event A which has probability strictly 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 conditional probabilities While these definitions are motivated by our intuitive notion of these concepts 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 wrong conclusions as illustrated for example by the various paradoxes in probability 1 Math 408 Actuarial Statistics I A J Hildebrand Independence is not the same as disjointness If A and B are disjoint corresponding 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 of the 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 and B are mutually exclusive that is if A occurs then B cannot occur and vice versa Independence and conditional probabilities in Venn diagrams In contrast to other properties such as disjointness independence can not be spotted in Venn diagrams On the other hand conditional probabilities have a natural interpretation in Venn diagrams The conditional probability given B is the probability you get if the underlying sample space S is shrunk to the set B i e everything outside B is deleted and then rescaled so as to have again unit area Verbal descriptions of conditional probabilities Whether a probability in a word problem represents a conditional or ordinary unconditional probability is not always obvious and you have to read the problem carefully to see which interpretation is the correct one Typically conditional probabilities are indicated by words like given if or among e g in the context of subpopulations though there are no 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 the entire problem Don t make assumptions about independence If a problem does not explicitly state that two events are independent they are probably not and not you should not make any assumptions about independence P A B is not the same as P B A In contrast to set theoretic operations like union or intersection in conditional probabilities the order of the sets matters P A B 0 is not the same as 1 P A B The complement formula only holds with respect to the first argument There is no corresponding formula for P A B 0 Independence of three or more events Though rarely needed the definition of independence of two events can be extended to three events as follows A B C are called mutually independent if the product formula holds for i the intersection of all three events i e P A B C P A P B P C and ii for any combination of 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 An are called independent if the product formula holds for any subcollection of these events 2
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