Bell Curve vs. Linear

Here’s a handy little chart showing the difference between a linear distribution like rolling a d20 and a bell curve distribution like 3d6 when it comes to rolling versus a target.  The first column is the d20 roll, the second is the approximate percent chance of rolling that or less on d20. That’s pretty obvious, but the next column is what the target number would be on 3d6 to have that chance to succeed (i.e. roll target or under).  So a 50% chance is right in the middle of the curve at 10… but by the time the target is 12 you’ve got a 75% chance of succeeding.  Next we have columns for a d20 skill roll/Basic Attack Bonus (as in 3e or 5e).  The final four columns show THAC0 (to hit AC 0) and what level you would have to be to have that chance of hitting an unarmored person, using the D&D Rules Cyclopedia as a reference point.  Hitting an unarmored person is the standard we’re using because that directly translates to scoring a hit in Heroes & Other Worlds/TFT (and similar games like Runequest) where armor reduces damage from a successful hit but does nothing to make success less likely.

From this you can see that, for instance, having a 13 DX in HOW is like being a 10th level Fighter, at least in terms of being able to land a blow.  (On the other hand, a 10th level Fighter in D&D can sustain multiple times the damage a HOW fighter could, so you can’t just translate back and forth quite that easily.) Another thing to pay attention to is the s20 skill column, where you can see that in terms of stat bonus, a D&D score of 18 is equivalent to DX 11 (if 18 is +3 as in original D&D), or maybe DX 12 (if 18 is +4 as in later editions).  Using the stat bonus in D&D is much more common than a straight roll-under against the stat.

Still, I find thinking of things this way as instructive.

Here’s an Anydice page with the 3d6 info, and just for the heck of it, the 1d20

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