Taming the bell | … making probability bell curves more ‘swing-ish’

I only realized something about multi-dice probability bell curves recently … (if you’re a maths-wiz, click away now as the following will probably only offend your sensibilities!).

Bad intuition
Instinctively, I believed that for example 2D6 has the same probability profile as D8+D4. That is, they both give numbers from 2 to 12 and each number on each die is just as likely to turn up.

But, I was quite wrong.

It turns out the bigger the separation in the die size, the “flatter” the probability bell curve becomes. That is, the more unpredictable the probability spread becomes. In retrospect, this makes sense.

What am I talking about and why could this be useful?
Below is an example where the number range 2 to 12 is generated using two dice, i.e. 2D6, D5+D7, D4+D8, D2+D10 and D1+D11:

2 to 12 ranges

In the Range 2 to 12 using two dice (made via anydice.com); the penultimate graph should have been labelled D2+D10 of course!

2 to 12 graph

As above, but presented graphically

So, it can be seen from the above that as the difference in the die size increases, the flatter the probability bell curve becomes (this means the probability curves is becoming more “swingish”).

This is another way to represent the ‘flattening’ of the bell curve; a D4+D8 has a much bigger flat top section (range 5-9) section than 2D6 (just for 7).

Range 2 to 12

Broader plateau seen in top D4+D8 vs bottom 2D6 covering the range 2-12 with two dice

Of course 2D6 systems are widespread in RPGs, for example see a recent post by Larry Hamilton concerning this: https://followmeanddie.com/2019/09/21/newfound-appreciation-of-the-2d6-table/

Here’s another example but for the range 3-18 using three dice (e.g. as used for PC stats, or the damage of a two-handed sword in AD&D).

3 to 18 range

In the Range 3 to 18 using three dice (made via anydice.com)

3 to 18 graph

As above, but presented graphically

So the message here is, when summing two (or more) dice, you can make the outcome more “random” (less constrained by the bell-shaped curve) by using a big spread in dice size. Edit: it has been pointed out to me that the term “random” as used here is better described mathematically as “variance”.

I think you mentioned useful?
In the main, this observation is not going to change your game. But, this phenomena is worth remembering for the old DM tool kit.  Below are some theoretical examples where this could come into play:

• Swap 2D6 random encounter tables for D8+D4 random encounter tables – instead of one monster being the most common (i.e. on a roll of 7), it’s a set of 5 most common monsters (i.e. rolls of 5-9 are equally common).

• Perhaps the fighter is drunk (and she is fighting with a two-handed sword; normally doing 3D6 damage), being drunk she is less in control of the weapon, and so is more unpredictable with it. Or, perhaps she is on her last hit-point and wants to try a ‘do-or-die’ attack. In either of the above cases, maybe the damage she does could be more unpredictable, more likely to be a flop or a whopper. So, to simulate either of the above, perhaps replace the normal (more predicable) 3D6 (3-18) with the more ‘swing-ish’ D10+2D4 (3-18) …

• 3D6 is also synonymous with rolling up PC stats e.g. STR, DEX, CON etc. But, if in your game you wanted more ‘swingy’ stats, then perhaps again consider using D10+2D4. Again, maybe let the player decide if they want to roll one or more ‘swingy stats’?

• 3D6 or 4D6 attribute tests – some gamers make ability checks using nD6 instead of a D20 (incidentally I compare the two systems here); if you wanted to make the nD6 test less predicable, use pairs of D4+D8 to replace pairs of 2D6s etc.

• Falling damage. Maybe falling down a large tree should be more unpredictable that falling off a cliff. Perhaps the branches might cushion the PC’s fall, or give the PC an extra thrashing on the way down?

• Perhaps you think fireball magic should be more unpredictable than is governed by standard bell curve generated by nD6. Again, mix it up.

• Recently, I’ve found a D6+D8 gives a better probability spread (for a Hex Flower I’m working on) than 2D7 would (which is lucky as I don’t own any D7s).

etc …

Take-home message
This observation is unlikely to change your game much, but ….

… if you get into the situation where a bunch of dice are being rolled and added, and you decide that you want that outcome to be less constrained by the bell, i.e. more “random”, then see if you can substitute the homogeneous dice with dice with bigger gaps between the dice size.

– – –

InHotS the cover imageMe on DriveThru; at the moment I’m mainly pimping my procedural:
:: High Seas ‘Hex Crawl’ – In the Heart of the Sea,
:: Wilderness Hex Crawl – In the Heart of the Unknown,
:: Dungeon/network generator – In the Heart of the Delve & Dangerous

11 thoughts on “Taming the bell | … making probability bell curves more ‘swing-ish’

  1. Max

    Very cool! I generally don’t even think about mixing dice of different size, but the joint probabilities are interesting. Intuitively it does make sense; any single die will have a uniform distribution, so the larger the difference in size, the less normalized the distribution will be because of the uneven weighting of the two dice on the joint probability. It’s almost like the Monty Hall Problem.

    It’s been a while since I’ve read it, and I’ve never played it, but you may want to check out the game Cryptomancer. I vaguely remember them using dice of different sizes in combined rolls in a way you rarely see in tabletop, and I remember thinking it may have an effect similar to what you’ve described here. Also, it’s just an interesting game in general if you’re a computationally-inclined person.

    Liked by 1 person

  2. Trent

    This is why in the AD&D Monster Manual II when they introduced random encounter tables based on 2-20, they said to roll 1d8+1d12 instead of 2d10: so the 5 results in the middle (9-13) are all equally likely to come up 🙂


    1. Goblin's Henchman Post author

      Hi Trent. Thanks for the comment.

      Interestingly, using D12+D8 (2-20) as mentioned in the MM2 came up on Reddit. On MM2 on Page 133, this D12+D8 system is mentioned without explanation of probability – thereafter there are just 2-20 lists (so if you missed the heading remark you’d naturally think to use 2D10). Had I noticed the leading remark (without explanation), I’d probably have still used 2D10.

      Later, on page 138 (after all the tables are done) an explanation is given. It’s a shame this explanation is tucked away – I wonder how many people missed this point. I certainly never noticed it before.


  3. Joel Priddy

    This seems especially useful for big pools of dice, where the bell curve becomes oppressively predictable. I’ll have to consider coming up with some handy dice recipes to replace, say, the fireball’s damage.

    Liked by 1 person

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  6. Jim

    This was really handy! I was just trying to think of a way to keep things relatively flat in a random table with 24 items, and I went with d20+d4 after reading this!


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