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Most any roll in Dungeon World that isn't a damage roll is 2d6 plus the relevant stat modifier. The results (barring explicit exceptions in the move) fall into the following three categories:
How likely is each outcome for the various modifiers?
The result of a roll is a bit more complicated than just 2d6+mod. Some moves, such as Aid, Bolster, and some item effects, grant +N forward or ongoing. Others, such as Interfere or Conditions, provide negative modifiers. That means the fixed range of stats modifiers (-3 to 3) isn't sufficient to show all possibilities in Dungeon World.
So, I've expanded the range to -5 to 5 for the following table. I've also included a column at @kviiri 's suggestion for the chance of 12+, which matters for triggering some moves.
\begin{array}{l|r|r|r|r|r} \text{Modifier} & \text{6 -} & \text{7 - 9} & \text{10 +} & &\text{12 +}\\ \hline \text -5 & 97.22\% & 2.77\% & 0.00\% & & 0.00\% \\ \text -4 & 91.67\% & 8.33\% & 0.00\% & & 0.00\% \\ \text -3 & 83.33\% & 16.67\% & 0.00\% & & 0.00\% \\ \text -2 & 72.22\% & 25.00\% & 2.77\% & & 0.00\% \\ \text -1 & 58.33\% & 33.33\% & 8.33\% & & 0.00\% \\ \text 0 & 41.67\% & 41.67\% & 16.67\% & & 2.77\% \\ \text 1 & 27.78\% & 44.44\% & 27.78\% & & 8.33\% \\ \text 2 & 16.67\% & 41.67\% & 41.67\% & & 16.67\% \\ \text 3 & 8.33\% & 33.33\% & 58.33\% & & 27.78\% \\ \text 4 & 2.77\% & 25.00\% & 72.22\% & & 41.67\% \\ \text 5 & 0.00\% & 16.67\% & 83.33\% & & 58.33\% \\ \end{array}
There's also an interesting turn of events here: Barbarians like to break things, including the 2d6 rule. They have two or more Appetites that allow/require them to instead roll 1d6 + 1d8. That results in the following table.
\begin{array}{l|r|r|r|r|r} \text{Modifier} & \text{6 -} & \text{7 - 9} & \text{10 +} & &\text{12 +} \\ \hline \text -5 & 87.50\% & 12.50\% & 0.00\% & & 0.00\% \\ \text -4 & 79.17\% & 18.75\% & 2.08\% & & 0.00\% \\ \text -3 & 68.75\% & 25.00\% & 6.25\% & & 0.00\% \\ \text -2 & 56.25\% & 31.25\% & 12.50\% & & 2.08\% \\ \text -1 & 43.75\% & 35.42\% & 20.83\% & & 6.25\% \\ \text 0 & 31.25\% & 37.50\% & 31.25\% & & 12.50\% \\ \text 1 & 20.83\% & 35.42\% & 43.75\% & & 20.83\% \\ \text 2 & 12.50\% & 31.25\% & 56.25\% & & 31.25\% \\ \text 3 & 6.25\% & 25.00\% & 68.75\% & & 43.75\% \\ \text 4 & 2.08\% & 18.75\% & 79.17\% & & 56.25\% \\ \text 5 & 0.00\% & 12.50\% & 87.50\% & & 68.75\% \\ \end{array}
Bonus: the Appetites move also has a clause that triggers when the d6 rolls higher than the d8. This chance is independent of the modifier, and is 31.25%
A nice way of factoring in variable modifiers into a probability calculation is modeling them as dice. The more probable a modifier is, the more frequent it appears on the model die's faces.
With Dungeon World, the most consistent modifier is the stat modifier arising from the character's stats. In a bird's eye view of the system, we can assume that one of the six stats of the character will be affecting any roll at any moment, so a six-sided die can be chosen to pick our random stat for the purposes of probability calculation.
In this AnyDice program, I have created such a die (named M for modifier) and mapped it into the standard range of stat modifiers assigned at character creation. There's 1/6 probability that your modifier will be a +2, 1/3 for +1, 1/3 for nothing and 1/6 for -1.
If we roll that die along with the usual 2d6, we get a bell curve distribution of 1 through 14. That output somewhat includes our answer but it's not clear enough. So I also made a function for sorting those results into the three categories of "clean hit", "hit with consequences" and "miss", represented as 1, 0, and -1 respectively.
So the final results are:
There are other occasional modifiers coming into play through other moves, but they are so much dependent on a specific game context that they would be very hard to determine how often they would come up. I chose to ignore them, but it is always possible to add more modifier dice into such a calculation.
