Here's a simple visual illustration.
There are 36 possible ways two rolls of a six-sider can come up:
(1,1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2,1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3,1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4,1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5,1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6,1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
Assuming that you're using fairly balanced d6s, all 36 of these results are equally probable. (You'll have to take my word for this part ā proving that the distinct compounds of equally probable events are also equally probable is beyond the scope of this post!)
Now here's the illustrative part.
Above, the pairs are sorted by whatever the first die rolled: all of the pairs that begin with a 1 are in the first row, all of the pairs that begin with a 2 are in the second row, and so forth.
What happens if we sort them by their sums instead?
(1, 1) = 2
(1, 2) (2, 1) = 3
(1, 3) (2, 2) (3, 1) = 4
(1, 4) (2, 3) (3, 2) (4, 1) = 5
(1, 5) (2, 4) (3, 3) (4, 2) (5, 1) = 6
(1, 6) (2, 5) (3, 4) (4, 3) (5, 2) (6, 1) = 7
(2, 6) (3, 5) (4, 4) (5, 3) (6, 2) = 8
(3, 6) (4, 5) (5, 4) (6, 3) = 9
(4, 6) (5, 5) (6, 4) = 10
(5, 6) (6, 5) = 11
(6, 6) = 12
See what's happening there? When sorted by their sums, those 36 equally probable results end up in eleven distinct buckets, and some of those buckets have more results in them than others.
To put it another way, you're six times as likely to roll a 7 as you are to roll a 2 because there are six times as many ways to roll a 7 than there are to roll a 2.
This V-shaped distribution holds true for all sums of two identical dice, with the peak at exactly (number of sides + 1).
(This is the central statistical trick behind the Apocalypse Engine, incidentally. Rolling flat nothing feels like it ought to be risky, but in fact you'll get a 7+ about three times in five, with 7 itself being particularly common. Under these conditions, rolling exactly 7 feels like you just barely made it, but it was in fact the single most likely outcome!)
germansingergirl
baratheonpilled
dailybridgerton
prguitarman
