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The faces of two regular dodecahedra are labeled with the numbers 1 to 12 in order to make dice. If these dice are rolled, what is the probability that the sum of the two top numbers is greater than \(18\)?

 Feb 8, 2022
 #1
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First we need to find a list of what numbers from both sets {1,2,3,4,5,6,7,8,9,10,11,12} and {1,2,3,4,5,6,7,8,9,10,11,12} will add up to 19 or greater. These are:

 

12+7=19

12+8=20

12+9=21

12+10=22

12+11=23

12+12=24

 

11+8=19

11+9=20

11+10=21

11+11=22

 

10+9=19

10+10=20

 

This amounts to 12 combinations. Since this can happen on opposite dice, we multiply by 2 to get 24 outcomes. The total number of outcomes are 12*12=144. So, the answer is 24/144 = 1/6.

 

Let me know if I did anything wrong!

 Feb 8, 2022
 #2
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You did everything right except when multiplied by 2. 3 combinations are considered one and the same:

 

10  +  10 ==20,   11  +  11 == 22,   12  +  12 ==24

 

Therefore, there are: 21 / 144 ==7 / 48 - which is the probability.

 Feb 8, 2022

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