Here are all the possible rolls and their sums. Count them. There are 36 of them.
(1,1, 1+1=2) (1,2, 1+2=3) (1,3, 1+3=4) (1,4, 1+4=5) (1,5, 1+5=6) (1,6, 1+6=7)
(2,1, 2+1=3) (2,2, 2+2=4) (2,3, 2+3=5) (2,4, 2+4=6) (2,5, 2+5=7) (2,6, 2+6=8)
(3,1, 3+1=4) (3,2, 3+2=5) (3,3, 3+3=6) (3,4, 3+4=7) (3,5, 3+5=8) (3,6, 3+6=9)
(4,1, 4+1=5) (4,2, 4+2=6) (4,3, 4+3=7) (4,4, 4+4=8) (4,5, 4+5=9) (4,6, 4+6=10)
(5,1, 5+1=6) (5,2, 5+2=7) (5,3, 5+3=8) (5,4, 5+4=9) (5,5, 5+5=10) (5,6, 5+6=11)
(6,1, 6+1=7) (6,2, 6+2=8) (6,3, 6+3=9) (6,4, 6+4=10) (6,5, 6+5=11) (6,6, 6+6=12)
For the probability that the sum is greater than 9, I will color the
ones that have sum greater than 9 red:
(1,1, 1+1=2) (1,2, 1+2=3) (1,3, 1+3=4) (1,4, 1+4=5) (1,5, 1+5=6) (1,6, 1+6=7)
(2,1, 2+1=3) (2,2, 2+2=4) (2,3, 2+3=5) (2,4, 2+4=6) (2,5, 2+5=7) (2,6, 2+6=8)
(3,1, 3+1=4) (3,2, 3+2=5) (3,3, 3+3=6) (3,4, 3+4=7) (3,5, 3+5=8) (3,6, 3+6=9)
(4,1, 4+1=5) (4,2, 4+2=6) (4,3, 4+3=7) (4,4, 4+4=8) (4,5, 4+5=9) (4,6, 4+6=10)
(5,1, 5+1=6) (5,2, 5+2=7) (5,3, 5+3=8) (5,4, 5+4=9) (5,5, 5+5=10) (5,6, 5+6=11)
(6,1, 6+1=7) (6,2, 6+2=8) (6,3, 6+3=9) (6,4, 6+4=10) (6,5, 6+5=11) (6,6, 6+6=12)
Count the red ones. There are 6 of them. that's 6 out of 36, so the
probability is the fraction with the number that have sum greater than
9 as the numerator and the total number of possibilities as the
denominator. That fraction is 6/36 which reduces to 1/6.
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For the probability that the sum is not divisible by 6, I will color the
ones with sums that are not divisible by 6 red (only 6 and 12 are divisible
by 6, so most of them are colored red):
(1,1, 1+1=2) (1,2, 1+2=3) (1,3, 1+3=4) (1,4, 1+4=5) (1,5, 1+5=6) (1,6, 1+6=7)
(2,1, 2+1=3) (2,2, 2+2=4) (2,3, 2+3=5) (2,4, 2+4=6) (2,5, 2+5=7) (2,6, 2+6=8)
(3,1, 3+1=4) (3,2, 3+2=5) (3,3, 3+3=6) (3,4, 3+4=7) (3,5, 3+5=8) (3,6, 3+6=9)
(4,1, 4+1=5) (4,2, 4+2=6) (4,3, 4+3=7) (4,4, 4+4=8) (4,5, 4+5=9) (4,6, 4+6=10)
(5,1, 5+1=6) (5,2, 5+2=7) (5,3, 5+3=8) (5,4, 5+4=9) (5,5, 5+5=10) (5,6, 5+6=11)
(6,1, 6+1=7) (6,2, 6+2=8) (6,3, 6+3=9) (6,4, 6+4=10) (6,5, 6+5=11) (1,5, 6+6=12)
Count the red ones. There are 30 of them. that's 30 out of 36, so the
probability is the fraction with the number that have sum not divisible
by 6 as the numerator and the total number of possibilities as the
denominator. That fraction is 30/36 which reduces to 5/6.
Edwin
