Explanation Formula used: If each simple event in sample space S is likely to occur as any other, then the probability of an arbitrary event E is given by, P ( E ) = Number of elements of E Number of elemenst of S = n ( E ) n ( S ) Theorem used: Bayes’ Formula: Let U 1 , U 2 , ... , U n be n mutually exclusive events whose union is the sample space S . Let E be an arbitrary event in S such that P ( E ) ≠ 0 . Then, P ( U 1 | E ) = P ( U 1 ∩ E ) P ( E ) = P ( U 1 ∩ E ) P ( U 1 ∩ E ) + P ( U 2 ∩ E ) + ... + P ( U n ∩ E ) = P ( E | U 1 ) P ( U 1 ) P ( E | U 1 ) P ( U 1 ) + P ( E | U 2 ) P ( U 2 ) + ... + P ( E | U n ) P ( U n ) Similar results hold for U 2 , U 3 , ... , U n . Calculation: Using Venn diagram: From the Venn diagram, the total number of trails is, n ( S ) = 15 + 25 + 35 + 25 = 100 The number of elements in R is, n ( R ) = 25 + 35 = 60 Also, n ( U 1 ∩ R ) = 25 and n ( U 2 ∩ R ) = 35 . The probability to be calculated is, P ( U 1 | R ) = P ( U 1 ∩ R ) P ( R )

College Mathematics for Business, Economics, Life... (Instructor's)

14th Edition

ISBN: 9780134676104

Author: Barnett

Publisher: PEARSON

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Chapter 8.4, Problem 13E

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