[RPG] Is 3d6 the same as 1d18


This might actually be a better fit for Mathematics SE, but since RPGs are the intended application of the information I've decided to put it here (it has nothing to do with me being lazy and not wanting to bother with the 15-second registration for a new SE, honest).

I've heard that, apparently, rolling three six-sided dice produces a bell-curve distribution, with the results weighted towards the middle (which would be 10.5). But I don't see how this works; if you have an equal probability of each result from one to six on each die, shouldn't you then have an equal probability of each result from three to eighteen from summing the results of three dice? If 1 and 6 are equally probable, aren't 1+1+1 and 6+6+6 also equally probable?

Best Answer

No, It's Not

You can use AnyDice to visualize dice rolls really easily and see what's going on. The links will show a table with the results for each one.

Here's 3d6.

Here's 1d18.

Here's both on the same screen for easy comparison.

Aside from the obvious issue of not being able to roll 1 on 3d6 because the minimum on each die is 1, the numbers in the middle will come up more often because there are more combinations that make them occur.

You're right, 1+1+1 and 6+6+6 are equally common, but there are 6^3 (216) possible permutations of results in 3d6 and only one of them comes out to 18. Quite a few of them come out to 9 though (6+1+2, 3+3+3, 4+2+3, etc). Comparatively on 1d18, there are 18 permutations and one of them is 18. Every number is equally likely. Thus, rolling 1d18 is going to be a lot more "swingy" (seeing the highs and the lows more often) than 3d6 will be. This has implications if you're making a dice rolling system, particularly if you have some kind of critical success or fail at the extremes. You'll see a minimum roll on 1d18 far more often than you'll see a minimum roll on 3d6.

Here's a good article explaining the math behind it. Any questions beyond that are probably better asked on Mathematics.SE. :)