\r\n" );
document.write( "Let the two integers picked be x and y.\r\n" );
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document.write( "First we'll determine the denominator of the desired probability, which\r\n" );
document.write( "is this set of points (x,y) so that
,
, \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
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document.write( "That is an array of 13x13 or 169 points (x,y). So the denominator of the\r\n" );
document.write( "desired probability is 169. \r\n" );
document.write( " \r\n" );
document.write( "Now let's find the numerator. We require that (x,y) be such that\r\n" );
document.write( " \r\n" );
document.write( "x + y < xy\r\n" );
document.write( " \r\n" );
document.write( "y - xy < -x\r\n" );
document.write( " \r\n" );
document.write( "y(1-x) < -x\r\n" );
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document.write( "Case 1: 1-x > 0\r\n" );
document.write( " -x > -1\r\n" );
document.write( " x < 1\r\n" );
document.write( "\r\n" );
document.write( "which of course means
\r\n" );
document.write( " \r\n" );
document.write( "Divide both sides of\r\n" );
document.write( "\r\n" );
document.write( "y(1-x) < -x\r\n" );
document.write( "\r\n" );
document.write( "by 1-x which is positive and doesn't reverse the <\r\n" );
document.write( " \r\n" );
document.write( "y <
\r\n" );
document.write( " \r\n" );
document.write( "y <
\r\n" );
document.write( " \r\n" );
document.write( "y <
\r\n" );
document.write( " \r\n" );
document.write( "y <
\r\n" );
document.write( " \r\n" );
document.write( "Let's draw the boundary curve y =
for the part involved\r\n" );
document.write( "in case 1: \r\n" );
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document.write( "\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "The solutions for case 1 are the 48 points indicated below the graph above\r\n" );
document.write( "They form a 7x7 array of points except for (0,0), which is on the graph.\r\n" );
document.write( "So that makes 49-1 or 48 points, marked above with dots.\r\n" );
document.write( " \r\n" );
document.write( "-------------\r\n" );
document.write( " \r\n" );
document.write( "Case 2: 1-x < 0\r\n" );
document.write( " -x < -1\r\n" );
document.write( " x > 1\r\n" );
document.write( " \r\n" );
document.write( "which of course means
\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides of\r\n" );
document.write( "\r\n" );
document.write( "y(1-x) < -x\r\n" );
document.write( "\r\n" );
document.write( "by 1-x which is negative and thus reverses the <\r\n" );
document.write( " \r\n" );
document.write( "y >
\r\n" );
document.write( " \r\n" );
document.write( "y >
\r\n" );
document.write( " \r\n" );
document.write( "y >
\r\n" );
document.write( " \r\n" );
document.write( "y >
\r\n" );
document.write( " \r\n" );
document.write( "Let's draw the boundary curve y =
for the part involved\r\n" );
document.write( "in case 2: \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "The solutions for case 2 are the 24 points indicated above the graph above.\r\n" );
document.write( "They form a 5x5 array of points except for (2,2), which is on the graph.\r\n" );
document.write( "So that makes 25-1 or 24 points marked above with dots.\r\n" );
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document.write( "That's a total of 48+24 = 72 points. \r\n" );
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document.write( "So the probability is
\r\n" );
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document.write( "To see all 72 points (x,y), let's graph the whole thing. You can see the 48 \r\n" );
document.write( "points (x,y) under the curve on the left, which came from case 1, and the 24\r\n" );
document.write( "points (x,y) above the curve on the right, which came from case 2, making 72\r\n" );
document.write( "in all:\r\n" );
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document.write( "
\r\n" );
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document.write( "Answer:
\r\n" );
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document.write( "Edwin
\n" );
document.write( "