document.write( "Question 1041294: If the sides of a triangle are decided by the throw of
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\n" ); document.write( "triangle is of maximum area given that it is an isosceles
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Algebra.Com's Answer #660591 by Edwin McCravy(20077)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The triangle has maximum area when all three dice are 6's.\r\n" );
document.write( "That's the case of the three rolls being (6,6,6).  So \r\n" );
document.write( "there is only one way the triangle can have maximum area.\r\n" );
document.write( "So the numerator of the probability will be 1.\r\n" );
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document.write( "Now we calculate the denominator of the probability.\r\n" );
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document.write( "Since we are given that the three dice rolls must result in an \r\n" );
document.write( "isosceles triangle, we must count the number of rolls that would \r\n" );
document.write( "result in such.  \r\n" );
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document.write( "In doing so, we must realize that rolls such as (2,2,6) do not \r\n" );
document.write( "form an isosceles triangle, even though two rolls are the same.  \r\n" );
document.write( "To form a triangle, the sum of the shortest two of the sides must \r\n" );
document.write( "be greater than the longest of the three sides.  That is not true \r\n" );
document.write( "in the case of rolls (2,2,6) since 2+2 = 4 < 6.\r\n" );
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document.write( "We calculate the number of rolls with two or three rolls the same\r\n" );
document.write( "which will produce an isosceles triangle.\r\n" );
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document.write( "Case 1:  the three rolls are equal resulting in an equilateral \r\n" );
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document.write( "There are obviously 6 of these: (1,1,1),(2,2,2),...,(6,6,6)\r\n" );
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document.write( "Case 2: Two of the sides are equal, and the third side not equal,\r\n" );
document.write( "resulting in a non-equilateral isosceles triangle.\r\n" );
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document.write( "First we find what three rolls result in a non-equilateral isosceles\r\n" );
document.write( "triangle, then we'll order them as to which of the three roll \r\n" );
document.write( "numbers 1st, 2nd, and 3rd they were obtained in.\r\n" );
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document.write( "Sub-case 1: the two equal rolls are 1's.\r\n" );
document.write( "There is no way to choose an unequal roll that will\r\n" );
document.write( "result in a triangle. [the case (1,1,1) is counted in case 1]\r\n" );
document.write( "That's 0 ways.\r\n" );
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document.write( "Sub-case 2: the two equal rolls are 2's.\r\n" );
document.write( "The unequal roll can only be 1 or 3 for it to\r\n" );
document.write( "result in a non-equilateral isosceles triangle.\r\n" );
document.write( "That's 2 ways.\r\n" );
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document.write( "Sub-case 3: the two equal rolls are 3's.\r\n" );
document.write( "The unequal roll can only be 1,2,4, or 5 for it to\r\n" );
document.write( "result in a non-equilateral isosceles triangle.\r\n" );
document.write( "That's 4 ways.\r\n" );
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document.write( "Sub-case 4: the two equal rolls are 4's.\r\n" );
document.write( "The unequal roll can only be 1,2,3,5, or 6 for it to\r\n" );
document.write( "result in a non-equilateral isosceles triangle.\r\n" );
document.write( "That's 5 ways.\r\n" );
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document.write( "Sub-case 5: the two equal rolls are 5's.\r\n" );
document.write( "The unequal roll can only be 1,2,3,4, or 6 for it to\r\n" );
document.write( "result in a non-equilateral isosceles triangle.\r\n" );
document.write( "That's 5 ways.\r\n" );
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document.write( "Sub-case 6: the two equal rolls are 5's.\r\n" );
document.write( "The unequal roll can only be 1,2,3,4 or 5 for it to\r\n" );
document.write( "result in a non-equilateral isosceles triangle.\r\n" );
document.write( "That's 5 ways.\r\n" );
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document.write( "So for case 2 there are 0+2+4+5+5+5 = 21 ways to choose\r\n" );
document.write( "the 3 rolls, unordered.\r\n" );
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document.write( "As we mentioned earlier, there are three \"roll numbers\", 1st roll, \r\n" );
document.write( "2nd roll and 3rd roll.\r\n" );
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document.write( "We must consider these orders for case 2 where there is one \r\n" );
document.write( "unequal roll.\r\n" );
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document.write( "We can choose the roll number for the unequal roll in 3 ways,\r\n" );
document.write( "and the equal pair of rolls will automatically be in the remaining\r\n" );
document.write( "two roll numbers.\r\n" );
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document.write( "So there are 3 ways to assign roll numbers to each of the 21\r\n" );
document.write( "cases of 2 equal rolls and 1 unequal roll that will result is\r\n" );
document.write( "a non-equilateral isosceles triangle.\r\n" );
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document.write( "That's 21*3 = 63 ways to choose the three rolls, where\r\n" );
document.write( "exactly two are equal.\r\n" );
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document.write( "So for cases 1 and 2 there are 6+63 = 69 rolls that will result\r\n" );
document.write( "in isosceles triangles.\r\n" );
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document.write( "So the desired conditional probability is 1/69.\r\n" );
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document.write( "Edwin
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