document.write( "Question 1137820: Suppose that a single fair die is rolled one time. Let events A and B be defined as:\r
\n" ); document.write( "\n" ); document.write( "A = { 1, 4, 5 } and B = { 2, 4, 5, 6 }\r
\n" ); document.write( "\n" ); document.write( "Are A and B independent events? Show why or why not? I cant figure this problem out
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Algebra.Com's Answer #755705 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
The intuitive method:\r\n" );
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document.write( "Two events A and B are independent of each other if and only if the probability\r\n" );
document.write( "of either event would not change if it were known that the other event has\r\n" );
document.write( "occurred.  That is:  P(A|B) = P(A) and P(B|A) = P(B)\r\n" );
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document.write( "P(A) = P( { 1, 4, 5} ) = 3/6 = 1/2,  \r\n" );
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document.write( "P(B) = P( {2, 4, 5, 6 }) = 4/6 = 2/3, \r\n" );
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document.write( "P(A & B) = P( {4} ) = 1/6\r\n" );
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document.write( "P(A|B) = P(A & B)/P(B) = (1/6)÷(2/3) = (1/6)(3/2) = 3/12 = 1/4\r\n" );
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document.write( "P(A|B) = 1/4 is not equal to P(A) = 1/2,\r\n" );
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document.write( "So since the knowledge of B's occurrence would cause the probability\r\n" );
document.write( "of A to increase from 1/4 to 1/2, they are not independent.\r\n" );
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document.write( "There is another way but it is not intuitive:\r\n" );
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document.write( "Two events A and B are independent if P(A & B) = P(A) × P(B)\r\n" );
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document.write( "P(A) = P( { 1, 4, 5} ) = 3/6 = 1/2,  \r\n" );
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document.write( "P(B) = P( {2, 4, 5, 6 }) = 4/6 = 2/3, \r\n" );
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document.write( "P(A & B) = P( {4} ) = 1/6\r\n" );
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document.write( "P(A) × P(B) = (1/2)(2/3) = 1/3\r\n" );
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document.write( "So they are not independent by the non-intuitive method.\r\n" );
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document.write( "Edwin

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