D&D General - Replacing 1d20 with 3d6 is nearly pointless

Mostly what it actually does is double bonuses and DCs, outside of crit hit/miss zones.

Change attribute bonuses to be (stat - 10), halved for damage. Make AC delta from 10 double -- so mage armor is 16+dex, plate is 26, shields are +4. Enchanted weapons go up to +6 to hit. Proficiency starts at +4 and scales up to +12 (expertise twice that).

Saves are 6+proficiency+attribute bonus. Bless is +1d8 instead of +1d4 (or 1d4*2). Bard inspiration dice are doubled. Etc.

You can see this graphically -- see AnyDice select "graph" and "at least" to see two curves right on top of each other -- but a concrete example might help.

Level 20 fighter with 24 strength attacking a foe in +3 plate and shield. (note that this isn't contrived -- I just picked out some reasonably extreme examples).

With 3d6, this is +13+3d6 vs AC 26. 25.93% chance of hitting.

Under the "double modifiers", this is 1d20+26 vs AC 42. 25% chance of hitting.

I got lucky -- I don't expect that to usually happen. So, I can do this again. A level 10 20 dex fighter with a +2 bow, +1 arrows and archery style using sharpshooter attacking a suit of animated armor (AC 18).

3d6 +5 (dex) +4(prof) +2 (bow) +1 (arrows) +2 (style) -5 (SS) = 3d6+9 vs AC 18. 74.07% chance of hitting.
1d20 + 10(dex) +8(prof) +4(bow) +2 (arrows) +4(style) -10 (SS)= 1d20+18 vs AC 26. 65% chance of hitting.

This one is about the peak of the difference you get.

Level 1 warlock with 16 charisma doing an EB on a dex 14 leather-clad bounty hunter.

3d6 + 3 (cha) +2 (prof) vs AC 13 = 90.74%
1d20 + 6 (cha) + 4 (prof) vs AC 16 = 85%

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Why this works is because the mean of 3d6 is 10.5, just like 1d20. The standard deviation of 1d20 is sqrt( 399/12 ) and the standard deviation of 3d6 is sqrt( 3 * 35/12 ) = sqrt( 105/12 ). Which means SD(1d20) = 2 * SD(3d6).

When we double the modifiers on 1d20 rolls, we in effect halve its standard distribution. The resulting distributions have the same first and second moments. The difference -- the third moment -- is far smaller than you'd naively expect from looking at the "roll exactly" curves.

D&D is a roll over/under system, not a roll exactly. Looking at "roll exactly" curves is looking at the derivative, when you should be looking at the values -- which is the cumulative distribution function. On anydice, this is "roll at least" or "roll at most" graphs.

Note that 2d10 is about half way, and represents about x1.5 modifiers.

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