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Suppose that three fair dice are thrown. Compute the probability of the following events:
\n" ); document.write( "(a) the event is ‘big’, that is, the sum of the three dice is at least 10 but
\n" ); document.write( "the three dice do not show the same value;
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document.write( "Hmm! Does this mean that none of the three are the same? Or can 2 be the same\r\n" );
document.write( "and the other different? I will assume none of the three are the same.\r\n" );
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document.write( "The largest roll must be 5 or 6, because 4+3+2 is only 9.\r\n" );
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document.write( "If the largest roll is a 6, the other two must be at least 4.\r\n" );
document.write( "The other two can be any of these 9 sets:\r\n" );
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document.write( "{1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5} \r\n" );
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document.write( "If the largest roll is a 5, the other two must be {1,4}, {2,3}, {2,4}, {3,4}\r\n" );
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document.write( "That's 13 sets of 3 rolls with sum at least 10 where all are different.\r\n" );
document.write( "There are 3! = 6 permutations of each.\r\n" );
document.write( "So there are 13*6 or 78 \"big\" rolls.\r\n" );
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document.write( "So the probability of \"big\" is 76/216 = 19/54\r\n" );
document.write( "\n" ); document.write( "(b) the event that the sum is 13;
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document.write( "There are only 2 sets of rolls that sum to 13. {6,5,2} and {6,4,3}\r\n" );
document.write( "Each can be permuted 3!=6 ways, so there are 2*6 = 12 rolls of 13. \r\n" );
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document.write( "So the probability is 12/216 = 1/18 \r\n" );
document.write( "\n" ); document.write( "(c) the event that a specific combination of two different numbers appears,
\n" ); document.write( "e.g. {3, 4} appears.
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document.write( "They are {3,4,1},{3,4,2},{3,4,3},{3,4,4},{3,4,5},{3,4,6}\r\n" );
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document.write( "4 of them are all different and can be permted 3! or 6 ways.\r\n" );
document.write( "That's 4*6 = 24 ways.\r\n" );
document.write( "The other 2, {3,4,3} and {3,4,4} can be permuted 3 ways each\r\n" );
document.write( "That's 2*3 = 6 ways.\r\n" );
document.write( "Total = 24+6 = 30\r\n" );
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document.write( "Probability = 30/216 = 5/36\r\n" );
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document.write( "Edwin \n" );
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