document.write( "Question 432850: Suppose we roll two ordinary, 6-sided dice. What is the expectation of the sum
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Algebra.Com's Answer #300055 by Edwin McCravy(20054)\"\" \"About 
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Suppose we roll two ordinary, 6-sided dice. What is the expectation of the sum
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document.write( "Expectation of sum:\r\n" );
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document.write( "Here are all 36 possible rolls with a pair of dice:\r\n" );
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document.write( "(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)\r\n" );
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document.write( "(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)\r\n" );
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document.write( "(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)\r\n" );
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document.write( "(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)\r\n" );
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document.write( "(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)\r\n" );
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document.write( "(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)\r\n" );
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document.write( "Their sums are\r\n" );
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document.write( "  2      3       4      5      6      7\r\n" );
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document.write( "  3      4       5      6      7      8\r\n" );
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document.write( "  4      5       6      7      8      9\r\n" );
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document.write( "  5      6       7      8      9     10\r\n" );
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document.write( "  6      7       8      9     10     11\r\n" );
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document.write( "  7      8       9     10     11     12\r\n" );
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document.write( "There is 1 2, 2 3's, 3 4's, 4 5's, 5 6's 6 7's, \r\n" );
document.write( "5 8's, 4 9's, 3 10's, 2 11's and 1 12.  And there'\r\n" );
document.write( "are 36 possible rolls, so the probabilities are the\r\n" );
document.write( "number of ways to roll the sum over 36.\r\n" );
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document.write( "So we list the probablity distribution function:\r\n" );
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document.write( "Sum of roll   Prob. of sum     \r\n" );
document.write( "    x            P(x)          x·P(x) \r\n" );
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document.write( "    2            1/36           2/36\r\n" );
document.write( "    3            2/36           6/36\r\n" );
document.write( "    4            3/36          12/36 \r\n" );
document.write( "    5            4/36          20/36\r\n" );
document.write( "    6            5/36          30/36\r\n" );
document.write( "    7            6/36          42/36\r\n" );
document.write( "    8            5/36          40/36\r\n" );
document.write( "    9            4/36          36/36\r\n" );
document.write( "   10            3/36          30/36 \r\n" );
document.write( "   11            2/36          22/36\r\n" );
document.write( "   12            1/36          12/36\r\n" );
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document.write( "TOTALS          36/36=1       252/36 = 7\r\n" );
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document.write( "Expectation = E(x) = ∑[x·P(x)] = 7\r\n" );
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document.write( "So if you rolled two dice many many times and averaged\r\n" );
document.write( "up all the sums, you would expect to get an average of about 7.\r\n" );
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document.write( "Expectation of maximum:\r\n" );
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document.write( "Here are all 36 possible rolls with a pair of dice:\r\n" );
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document.write( "(1,1)  (1,2)   (1,3)  (1,4)   (1,5)  (1,6)\r\n" );
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document.write( "(2,1)  (2,2)   (2,3)  (2,4)   (2,5)  (2,6)\r\n" );
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document.write( "(3,1)  (3,2)   (3,3)  (3,4)   (3,5)  (3,6)\r\n" );
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document.write( "(4,1)  (4,2)   (4,3)  (4,4)   (4,5)  (4,6)\r\n" );
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document.write( "(5,1)  (5,2)   (5,3)  (5,4)   (5,5)  (5,6)\r\n" );
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document.write( "(6,1)  (6,2)   (6,3)  (6,4)   (6,5)  (6,6)\r\n" );
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document.write( "The maximums are\r\n" );
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document.write( "  1      2       3      4      5      6\r\n" );
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document.write( "  2      2       3      4      5      6\r\n" );
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document.write( "  3      3       3      4      5      6\r\n" );
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document.write( "  4      4       4      4      5      6\r\n" );
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document.write( "  5      5       5      5      5      6\r\n" );
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document.write( "  6      6       6      6      6      6      \r\n" );
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document.write( "There is 1 1, 3 2's, 5 3's, 7 4's, 9 5's and 11 6's.  \r\n" );
document.write( "And there are 36 possible rolls, so the probabilities are the\r\n" );
document.write( "number of ways to roll the maximum over 36.\r\n" );
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document.write( "So we list the probablity distribution function:\r\n" );
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document.write( "Max of roll   Prob. of max     \r\n" );
document.write( "    x            P(x)          x·P(x) \r\n" );
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document.write( "    1            1/36           1/36\r\n" );
document.write( "    2            3/36           6/36\r\n" );
document.write( "    3            5/36          15/36 \r\n" );
document.write( "    4            7/36          28/36\r\n" );
document.write( "    5            9/36          45/36\r\n" );
document.write( "    6           11/36          66/36\r\n" );
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document.write( "TOTALS          36/36=1       161/36 = 4.47222···\r\n" );
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document.write( "Expectation = E(x) = ∑[x·P(x)] = 161/36\r\n" );
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document.write( "So if you rolled two dice many many times and averaged\r\n" );
document.write( "up all the maximums, you would expect to get an average of about 4.47222···\r\n" );
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document.write( "Edwin
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