Math 408 Actuarial Statistics I A J Hildebrand General Probability III Bayes Rule Bayes Rule 1 Partitions A collection of sets B1 B2 Bn is said to partition the sample space if the sets i are mutually disjoint and ii have as union the entire sample space A simple example of a partition is given by a set B together with its complement B 0 2 Total Probability Rule Average Rule If B1 B2 Bn partition the sample space then for any set A P A P A B1 P B1 P A Bn P Bn 1 3 Bayes Rule general case If B1 B2 Bn partition the sample space then for each i 1 2 n and any set A P A Bi P Bi P A Bi P Bi 2 P Bi A P A B1 P B1 P A Bn P Bn P A 4 Bayes Rule special case P A B P B P B A P A B P B P A B 0 P B 0 P A B P B P A 3 This corresponds to the choice B1 B B2 B 0 in the general case of Bayes Rule Notes and tips Memorizing Bayes Rule The Total Probability Rule says that the expression appearing in the denominator in Bayes Rule is equal to P A If you remember this rule you could get by memorizing the simpler version of Bayes Rule given by the latter formulas in parentheses in 2 and 3 However I recommend memorizing Bayes Rule in the first form since that is the form that you normally need in applications General versus special case of Bayes Rule Many but not all applications of Bayes Rule involve only the special case when the simpler formula 3 can be used However for more general problems one does need the more complicated formula 2 I recommend to memorize both formulas Interpretation of Bayes Rule Bayes Rule can be interpreted in terms of prior and posterior probabilities The prior probabilities are P Bi i e the ordinary probability that the event Bi occurs Bayes Rule shows how these probabilities change if we know that event A has occurred namely it gives a formula for P Bi A the conditional probability that Bi occurs given that A has occurred The latter probabilities are called posterior probabilities The terms prior and posterior come from Latin and mean before and after 1 Math 408 Actuarial Statistics I A J Hildebrand Recognizing Bayes Rule problems Bayes Rule is a formula for reversing the order in conditional probabilities Many but not all conditional probability problems in the actuarial exams are of this type If the probability sought in the problem is a conditional probability and the same conditional probability but with the order of events reversed is given or can easily be deduced from the given information the problem is likely a Bayes Rule problem Example In the drug test problem the probability sought is that of someone taking drugs given that he she tests positive whereas the reverse conditional probability that someone tests positive given that he she takes drugs is given Recognizing conditional probabilities Conditional probabilities are often indicated by words phrases like given that or if or by words implying a subpopulation Here are some examples of statements mostly taken from actuarial exam problems that refer to a conditional probability along with their translation into mathematical language The give away words that indicate that a conditional probability is involved are set in italics 5 percent of those taking drugs test negative Translation P test negative take drugs 0 05 For each smoker the probability of dying during the year is 0 05 Translation P dying smoker 0 05 A blood test indicates the presence of a disease 95 of the time the disease is actually present Translation P test indicates disease has disease 0 95 Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation problem Translation P smoker circulation problem 2 P smoker no circulation problem 2
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