document.write( "Question 1024795: Please please please help
\n" ); document.write( "Suppose E,F are events in a sample space S. Suppose Pr (E|F)=3/4 and Pr (F|E)= 1/2 and Pr (EUF)=2/3. Find (E) and Pr (F) \n" ); document.write( "
Algebra.Com's Answer #640194 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "There are two ways to do it;\r\n" );
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document.write( "(1) By Venn diagram\r\n" );
document.write( "(2) By formula\r\n" );
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document.write( "I'll do it both ways:\r\n" );
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document.write( "(1) By Venn diagram:\r\n" );
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document.write( "Let the probabilities of the four regions in the Venn\r\n" );
document.write( "diagram be a,b,c, and d: \r\n" );
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document.write( "Then if we understand what \r\n" );
document.write( "\"one set given another\" means, we have:\r\n" );
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document.write( "P(E|F) = b/(b+c) = 3/4\r\n" );
document.write( "P(F|E) = b/(a+b) = 1/2\r\n" );
document.write( "P(EUF) = (a+b+c)/(a+b+c+d) = 2/3\r\n" );
document.write( "P(U) = a+b+c+d = 1\r\n" );
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document.write( "Cross multiplying and simplifying on the first three,\r\n" );
document.write( "we have the system:\r\n" );
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document.write( "Subtracting the 3rd equation from the 4th,\r\n" );
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document.write( " d = 1-2d\r\n" );
document.write( "3d = 1\r\n" );
document.write( " d = 1/3\r\n" );
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document.write( "Since b=3c, c=b/3\r\n" );
document.write( "and we have b=a\r\n" );
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document.write( "Substitute those and d = 1/3 in\r\n" );
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document.write( "a+b+c=2d\r\n" );
document.write( "b+b+b/3 = 2(1/3)\r\n" );
document.write( "2b+b/3 = 2/3\r\n" );
document.write( "Multiply through by 3\r\n" );
document.write( "6b+b = 2\r\n" );
document.write( "7b = 2\r\n" );
document.write( "  b = 2/7\r\n" );
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document.write( "a = b = 2/7\r\n" );
document.write( "c = b/3 = (2/7)/3 = 2/21\r\n" );
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document.write( "So Pr(E) = a+b = 2/7+2/7 = 4/7\r\n" );
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document.write( "and Pr(F) = b+c = 2/7+2/21 = 8/21\r\n" );
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document.write( "By formulas, which requires no understanding:\r\n" );
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document.write( "Pr(E|F) = Pr(E&F)/Pr(F) = 3/4\r\n" );
document.write( "Pr(F|E) = Pr(F&E)/Pr(E) = 1/2\r\n" );
document.write( "Pr(EUF) = Pr(E)+Pr(F)-Pr(E&F) = 2/3\r\n" );
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document.write( "From the first:  3Pr(F) = 4Pr(E&F)\r\n" );
document.write( "                 Pr(F) = 4Pr(E&F)/3\r\n" );
document.write( "From the second: Pr(E) = 2Pr(F&E)\r\n" );
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document.write( "Substituting in:\r\n" );
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document.write( "Pr(E)+Pr(F)-Pr(E&F) = 2/3\r\n" );
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document.write( "2Pr(F&E)+4Pr(E&F)/3-Pr(E&F) = 2/3\r\n" );
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document.write( "Since F&E = E&F\r\n" );
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document.write( "2Pr(E&F)+4Pr(E&F)/3-Pr(E&F) = 2/3\r\n" );
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document.write( "Pr(E&F)+4Pr(E&F)/3 = 2/3\r\n" );
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document.write( "Multiply through by 3\r\n" );
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document.write( "3Pr(E&F)+4Pr(E&F) = 2\r\n" );
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document.write( "7Pr(E&F) = 2\r\n" );
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document.write( "Pr(E&F) = 2/7\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "Pr(E) = 2Pr(F&E)\r\n" );
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document.write( "Pr(E) = 2Pr(E&F)\r\n" );
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document.write( "Pr(E) = 2(2/7)\r\n" );
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document.write( "Pr(E) = 4/7\r\n" );
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document.write( "Pr(F) = 4Pr(E&F)/3\r\n" );
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document.write( "Pr(F) = 4(2/7)/3\r\n" );
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document.write( "Pr(F) = (8/7)/3\r\n" );
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document.write( "Pr(F) = 8/21\r\n" );
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document.write( "Edwin
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