Handout_5 General addition and multiplication rules

Handout_5 General addition and multiplication rules - Feb...

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Feb 21’10 GENERAL ADDITION AND MULTIPLICATION RULES, #5 CONDITIONAL PROBABILITY, AND BAYES’ THEOREM General Addition (Union) Rule . For two given events, A and B , the probability that event A , or event B , or both occur is equal to the probability that event A occurs, plus the probability that event B occurs, minus the probability that the events occur together. This addition rule may be written as P(A U B) = P(A) + P(B) – P(A B) (5.1) For mutually exclusive events A and B the probability P(A and B) = P(A B) = 0 and the rule (5.1) has a simpler form P(A U B) = P(A) + P(B) (see formula ( 4 .5) in handout # 4 ). /* The formula ( 4 .5) can be expanded. In the case of k mutually exclusive events E 1 , E 2 , …, E k the probability of occurrence of any of them (we treat it as an event E ) is k P(E) = P ( E 1 U E 2 U E 3 U …U E k ) = Σ P(E i ) (5.2) i =1 The relation (5.2) is known as the theorem of addition of probabilities . An event E defined in accordance with (5.2) is called the sum of events E 1 + E 2 +…+ E k . In a special case, when the events E 1 , E 2 , …, E k constitute a whole sample space for some activity, the sum (5.2) is equal to 1 due to the property of exhaustiveness ( 4 .4). */ We should apply the general addition rule (5.1) instead of ( 4 .5) when the events A and B under consideration are not mutually exclusive. In such a case, two sample spaces S A and S B associated with these events intersect . In other words, two sets of sample points (representing the outcomes of an experiment) related to the events A and B have at least one common sample point. Let us illustrate the idea with the following thought experiment_1 : if we roll a single die then the event A of showing an odd number and the event B of showing a number greater than 3 are not mutually exclusive. Such an event A can be realized in three ways: {1}, {3}, {5}; in other words, the sample space S A = {1, 3, 5 } . Similarly, the sample space S B = {4, 5 , 6} . We see that two sample spaces S A and S B intersect, and the outcome { 5 } is the common sample point. The situation may be illustrated also graphically, by means of a corresponding Venn diagram . Venn diagrams are simple “area” or “region” diagrams which are used to illustrate the relationship between different events . Usually they look like two or more circles, each including a set of sample points (sample space) related to one of the events under consideration. If the events do not have any point in common then the circles do not intersect; that is, the events under consideration are mutually exclusive. But if the
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This note was uploaded on 11/24/2011 for the course MATH 3000 taught by Professor Kzaer during the Spring '05 term at St. Johns College MD.

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Handout_5 General addition and multiplication rules - Feb...

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