Many games based on (or “powered by”) the Apocalypse engine use a core 2d6 mechanic: roll two six-sided dice and potentially add a modifier (usually between -1 to +3). If the result is 10 or higher, you succeed; if it’s 6 or less, the GM (“MC” in many games) gets to make a hard move; if it’s in between, you succeed but only partially or at a cost.
I wanted to take some time to examine the importance of a +1/2/3, as most situations don’t involve modifiers higher than that. Note that a +3 can take a 4 to a 7 or a 7 to a 10, so a lot of folks try to reach that between stat modifiers and potentially assistance from other players. Unlike the single d20 roll many people know from games like D&D, each potential result has a different probability, so let’s build a chart of each combination.
Visually, we can already evaluate some things. On a straight roll with no modifiers, you have slightly more than 1/2 chance of at least mixed success (green to purple). Every time you add or subtract one, just look at the color in that direction, so a +1 succeeds on the yellow line, etc.
Let me take a moment to talk about a common misunderstanding of probability. Imagine you have a 25% chance of success at something. That means that, on average, about 1 out of 4 attempts will succeed. Now imagine your chance increases to 50%. That’s not a “25% increase” – that’s a 100% increase, because you have doubled your chances: 2 out of 4 attempts. That means that, when your chances are small, even a minor bump can have a big effect.
Then I calculated the counts for modifiers up to +3 (the full chart is at the end of this article). Note that a +1 is a 2/3 increase in the chance for a full success, as it includes an extra 4 possible rolls more than the 6 you already had. It also results in about a 1/4 increase in the chance for at least some success – this is huge and possibly a great reason for an “aid” move (called “Aid or Interfere” in Dungeon World, but many systems have something similar).
if you have +2 in your main stat, then you have 5/6 chance of at least some success, and you have 2.5x the chance of a full success over a straight roll! Once you get to +3, you’re over 90% to succeed, most likely without a hitch.
Analysis of -1 is left as an exercise for the reader.
Finally, just a general game note: remember that a miss is not a “failure” – it just means the GM / MC gets to “make a move as hard as they like,” which can be !!FUN!!