Algebra.Com's Answer #822638 by ikleyn(52787)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Assume Pr[A]=0.3, Pr[B]=0.4, and Pr[A∩B]=0.05. Find the following values:
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document.write( "(a) Pr[B|A] =
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document.write( "(b) Pr[A|B′] =
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document.write( "(c) Pr[B′|A′] =
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document.write( " In this problem, all questions ask about CONDITIONAL PROBABILITY.\r
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document.write( " To solve the problem and to answer all the questions, you need firmly know WHAT the conditional probability is.\r
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document.write( " If X and Y are the events in the universal set of events, then P(X|Y) is THIS ratio\r
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document.write( " P(X|Y) =  . \r\n" );
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document.write( " Having this definition, let answer all questions, one after other.\r
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document.write( "(a) P(B|A) =  =  =  =  . ANSWER\r\n" );
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document.write( " Solved. So, it is easy.\r\n" );
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document.write( "(b) P(A|B').\r\n" );
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document.write( " We should find P(A ∩ B'). The set (A ∩ B') are those elements/(events) of A that do not belong to B.\r\n" );
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document.write( " From it, it is clear that the set (A ∩ B') is (A \ (A ∩ B)). \r\n" );
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document.write( " Therefore, P(A ∩ B') = P(A) - P(A ∩ B) = 0.3 - 0.05 = 0.25.\r\n" );
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document.write( " Next, P(B') = 1 - P(B) = 1 - 0.4 = 0.6.\r\n" );
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document.write( " THEREFORE, P(A|B') =  =  =  . ANSWER\r\n" );
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document.write( "(c) P(A'|B').\r\n" );
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document.write( " We should find P(A' ∩ B'). The set (A' ∩ B') are those elements/(events) of the universal set that belong NEITHER A NOR B.\r\n" );
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document.write( " From it, it is clear that the set (A' ∩ B') is the complement of (A U B). \r\n" );
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document.write( " P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.3 + 0.4 - 0.05 = 0.65.\r\n" );
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document.write( " Therefore, P(A' ∩ B') = 1 - P(A U B) = 1 - 0.65 = 0.35.\r\n" );
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document.write( " Next, P(B') = 1 - P(B) = 1 - 0.4 = 0.6.\r\n" );
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document.write( " THEREFORE, P(A'|B') =  =  =  . ANSWER\r\n" );
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document.write( "Solved.\r
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document.write( "All questions are answered.\r
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