document.write( "Question 1062142: Three fair six-sided number cubes, each with faces labeled 0,2,4,6,8,10 are all rolled. What is the probability that the sum of the numbers rolled is greater than 20? Express your answer with a common fraction. \n" ); document.write( "

Algebra.Com's Answer #676968 by KMST(5328) $\"\"$ $\"About$

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Same answer, different way of counting.
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\n" ); document.write( "There are $\"6\"$ possible outcomes for each of the number cubes,
\n" ); document.write( "so that makes $\"6%5E3=216\"$ possible and equally probable outcomes.
\n" ); document.write( "If we can tell the number cubes apart
\n" ); document.write( "(for example because they are different colors, or because we roll them separately), we can tell apart those 216 outcomes.
\n" ); document.write( "We would count 10, 10, and 8 in the red, white, and blue cubes respectively as one outcome,
\n" ); document.write( "10, 8, and 10 in the red, white, and blue cubes respectively as another one of the 216 equally probable outcomes,
\n" ); document.write( "and 8, 10, and 10 in the red, white, and blue cubes respectively as yet another one.
\n" ); document.write( "Of course, if the 3 cubes are identical, and they are rolled together, we could not distinguish those 3 outcomes, and we would just see that getting one 8 and two 10's is 3 times more likely than getting three 10's.\r
\n" ); document.write( "\n" ); document.write( "There is only way for the numbers on all 3 cubes to be 10,
\n" ); document.write( "but there would be $\"3\"$ ways to get two 10's and one 8,
\n" ); document.write( "each way having the 8 appear on a different cube.
\n" ); document.write( "For a set of 3 different numbers, such as 10, 8 and 6,
\n" ); document.write( "there are $\"3%21=1%2A2%2A3=6\"$ permutations that would give us that set of numbers.
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\n" ); document.write( "That said, what sums of numbers would be greater than 20?
\n" ); document.write( "We have
\n" ); document.write( " $\"10%2B10%2B10=30\"$ , $\"8%2B8%2B8=24\"$ ,
\n" ); document.write( "which can happen only one way each,
\n" ); document.write( "accounting for $\"2\"$ of the $\"216\"$ possible outcomes.
\n" ); document.write( "There are also $\"7\"$ sums greater than 20, that can happen $\"3\"$ different ways:
\n" ); document.write( " $\"10%2B10%2B8=28\"$ ,
\n" ); document.write( " $\"10%2B10%2B6=26\"$ ,
\n" ); document.write( " $\"10%2B10%2B4=24\"$ ,
\n" ); document.write( " $\"10%2B10%2B2=22\"$ ,
\n" ); document.write( " $\"10%2B8%2B8=26\"$ ,
\n" ); document.write( " $\"10%2B6%2B6=22\"$ , and
\n" ); document.write( " $\"8%2B8%2B6=22\"$ .
\n" ); document.write( "Those account for $\"7%2A3=red%2821%29\"$ of the $\"216\"$ outcomes.
\n" ); document.write( "Finally, there are $\"2\"$ sums greater than 20, that can happen $\"6\"$ different ways:
\n" ); document.write( " $\"10%2B8%2B6=24\"$ , and $\"10%2B8%2B4=22\"$ .
\n" ); document.write( "Those sums account for $\"2%2A6=green%2812%29\"$ of the $\"216\"$ possible outcomes.
\n" ); document.write( "All the other possible sums are equal or less than 20/.
\n" ); document.write( "The number of possible outcomes with a sum greater than 20 are
\n" ); document.write( " $\"2%2Bred%2821%29%2Bgreen%2812%29=35\"$ of the $\"216\"$ equally probable outcomes.
\n" ); document.write( "so, the probability that the sum of the numbers rolled is greater than 20 is
\n" ); document.write( " $\"highlight%2835%2F216%29\"$ .
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