# D&D GeneralReplacing 1d20 with 3d6 is nearly pointless

#### Esker

##### Hero
It occurs to me that one reason the centering and scaling doesn't match probabilities even better than it does is that the discrete nature of die rolls prevents us from actually matching the moments. But this could be dealt with by expanding the DCs in the modified d20 system around 11 instead of 10, and requiring a 50/50 "confirmation" roll if your roll exactly ties the DC. That essentially raises the DC by 1/2, as if you had expanded around 10.5.

#### Ovinomancer

##### No flips for you!
I'm not sure why you're focusing on the difference in spacing... In 5e's d20 system, the only thing that's relevant is your chances of meeting or exceeding some threshold; it doesn't matter at all how likely you are to roll any specific value, X, except insofar as that represents the difference in difficulty between a DC X and a DC X-1 roll.
I focus on it because it's a key indicator that you aren't comparing the same things. Here's a great example using the OP's first example. I'll repost it for ease:

OP example said:
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.

Alright, this is a true statement -- these things have similar probabilities (~1% different). But, what happens if we alter this highly contrived example just a touch? Let's give the Fighter STR 20, and the target a ring of protection. This makes the normal attack bonus +11 against normal AC 27. Need a 16+ on 3d6, or a 4.64% chance of success.

Now, if we do the doubling, it's an attack bonus of +22 (4 less) against an AC of 44 (2 more). You need a 22 on a d20, for a 0% chance of success.

Before you start, sure, 5e has a 20 always hits rule, but that's not part of the distribution -- it's actually breaking the statistical analysis -- so it cannot be used to justify a system based on an incorrect assumption of similarity in statistical behavior.

The difference in spacing is a critical indicator that such breakpoints exist, this one just barely off of the example put forth to show just how similar the systems are. If you, instead, reduce the difference by 2 instead of increase it, the delta is on 3d6 a 50% chance and on the doubled d20 a 45% chance. Reduce another 2 and it's 74% to 65%. You have a non-linear rate of change, which makes it obvious these two things are not, at all, alike.

The OP's suggestion was that using 3d6 is similar to a system where where bonuses are doubled, DCs (and similarly, ACs) are transformed to be DC' = 10 + 2*(DC - 10), and we use a d20 to resolve outcomes.

Note that this is the same in practice as taking the new DC to be 10.5 + 2*(DC-10.5), where we've used the actual expected value of the 1d20 and 3d6 rolls, because this is 0.5 lower than 10+2*(DC-10), and so it yields success on the same integers.

If you need a natural X to succeed in the 3d6 system (that is, you have a +Y bonus and the DC is X+Y), that becomes a +2Y bonus and a DC of 10+2*(X-10)+2Y. So you need a natural 10+2*(X-10), or 2*X - 10 in the modified d20 system. So we could compare success rates for each value of X with the corresponding target natural rolls.
Math tip: that equation simplifies to 2*DC-10. Much easier to write. It's also slightly off from what the OP suggests, but pretty close, for AC. For DC, it's more (DC-10)*2+6, or 2*DC-14. Still not quite what is suggested, but pretty close.

The OP actually suggests using ability-10 as the bonus rather than 2*ability bonus, which increments in possible single steps rather than by 2's all the time. Not sure why they felt the need for that bit of granularity while just doubling everything else, but hey, it's good.

I do find it odd that you think I misunderstood what the OP was suggesting. I didn't. It's just not based on good math.

Alternatively we could leave bonuses and DCs the same and scale and shift the roll instead. Compare 1d20 to 2*3d6-10. Using a target of X on the 2*3d6-10 roll is equivalent to a target of ... 2*X-10 on the modified d20, just the same as if we'd rescaled bonuses and DCs.
This does pretty much nothing for increasing correctness, but it does immediately show how sensitive the analysis is to the arbitrary selection of offset. Your suggestion now only lines up at 2, where the graphs cross, and then almost again at 13 and 14 on 3d6 (which get close to 15 and 17, respectively, on the d20).
So the OP was off by one in their graph in terms of illustrating the impact of their proposed system relative to using 3d6 with regular bonuses and DCs. But actually using -10 vs -11 only affects which system makes for easier rolls, not the sizes of the gaps, since 10 and 11 are equal distances from the mean, and so you're essentially just swapping successes and failures and inverting the labels on the x-axis.
No, the OP wasn't "off by one" as that actually makes the artificial comparison strictly worse. You'd need to choose a different scalar to align at 11 rather than 10. Which, interestingly, while the OP chose 10 as the center for the scaled 3d6, he left 10.5 for the d20 center, meaning the graphs don't even have the same mean value. The warning signs for bad math are just all over this -- and, indeed, it's terrible math.

#### Ovinomancer

##### No flips for you!
It occurs to me that one reason the centering and scaling doesn't match probabilities even better than it does is that the discrete nature of die rolls prevents us from actually matching the moments. But this could be dealt with by expanding the DCs in the modified d20 system around 11 instead of 10, and requiring a 50/50 "confirmation" roll if your roll exactly ties the DC. That essentially raises the DC by 1/2, as if you had expanded around 10.5.
Nope. Makes it worse. It might be nice if you at least plugged these things into Anydice or did some actual looking at what monstrosities you're birthing before throwing more bad math at bad math as if that'll fix it.

Using the OP example and the ones above, just reduce the likelihood of the d20 system hitting by 5% for each, making the deltas 5.93% on the OP (3d6 better), 10% on the next to last, and 19% on the last. The second one doesn't change -- 3d6 can still hit and d20 cannot.

And, one more observation on that example that d20 can't even hit -- 3d6 only goes up to 18, so d20 has 2 more numbers in it's upper range and still fails to be able to connect.

#### Esker

##### Hero
Nope. Makes it worse. It might be nice if you at least plugged these things into Anydice or did some actual looking at what monstrosities you're birthing before throwing more bad math at bad math as if that'll fix it.

The issue was that 2*3d6-11 has a mean of 10, whereas 1d20 has a mean of 10.5. By introducing a "confirmation" mechanic, we're effectively reducing every roll by 1/2 (if it's easier to see, you can imagine rolling the 1d2 confirmation die after every roll and subtracting 1 on a 1, and leaving the roll alone on a 2, but this has no impact unless our roll exactly meets the DC so in practice you'd only need to roll the confirmation die in that case).

This isn't reducing the likelihood to succeed by 5%; if applied to a d20 it's essentially reducing it by 2.5%, so it's as though we made the d20 have a mean of 10, like the 2*3d6-11.

But that's not actually quite want we want --- I did mess that up in hastily firing off an idea that I hadn't had a chance to sit down and work through yet. What we want is to apply the confirmation die in the 3d6 case, not the d20 case. We either (1) roll 2*3d6-10, applying the confirmation mechanic, which should be comparable to an unaltered d20 system, or (2) just roll 3d6 with the confirmation mechanic, and compare that to a d20 system where bonuses are doubled together with DCs' distance from 10.

Let's look at (1) first. Here are the probabilities of meeting each target natural roll from 1 to 20, comparing vanilla 1d20 (no changes to DCs or modifiers), compared to 2*3d6-10 with the confirmation correction:

Once we correct for the mean of 2*3d6-10 being off by a half, the approximation error is symmetric around the DC/bonus combinations with a 50% success rate. The 3d6 curve is of course nonlinear -- no points for pointing that out; everyone already recognized that -- and so the discrepancy in success rates is not constant, but it is at worst 4%.

And here are the probabilities for a "reduced randomness" scheme: 3d6 with a confirmation correction and no change to DCs or modifiers, compared to 1d20 with doubled modifiers and new DCs set to 10 + 2*(DC - 10) -- that is to say, expanding around 10 by a factor of 2. (The x-axis is the target 3d6 roll; the roll needed on the d20 is different)

As advertised, using 3d6 produces less variation in rolls. But if we adjust modifers and DCs we achieve the same effect with a d20. We do have a bit of an issue on the extremes, in that the scaled d20 approximation makes some rolls guaranteed or impossible, whereas they are around 96% or 4% with 3d6. But again that's something everyone has acknowledged throughout this thread, so there's nothing special in you pointing that out -- it's not a problem with anyone's math, it's a fact that distributional approximations like this have the worst fit in the tails.

Note that in the latter case, the curves cross at 10.5, not at 11. This is due to the fact that by altering our DCs, tasks that previously had DC 11+modifier are no longer 50% success rate tasks; they've gotten a little more difficult, whereas the 50% success rate now sits between old DC 10+modifier (still DC 10+modifier) and DC 11+modifier (now DC 12+modifier). Introducing the confirmation mechanic on the 3d6 roll does the same thing: you no longer necessarily succeed on a DC 11+modifier task if you roll a natural 11; you only have a 50% chance.

#### Ovinomancer

##### No flips for you!
The issue was that 2*3d6-11 has a mean of 10, whereas 1d20 has a mean of 10.5. By introducing a "confirmation" mechanic, we're effectively reducing every roll by 1/2 (if it's easier to see, you can imagine rolling the 1d2 confirmation die after every roll and subtracting 1 on a 1, and leaving the roll alone on a 2, but this has no impact unless our roll exactly meets the DC so in practice you'd only need to roll the confirmation die in that case).
Not what you say above, and still a really weird, non-mathematical thing to do to correct at a single point in the distribution because your choice of arbitrary centering (to get, recall, a optical match but not actual match in probabilities) to look better. You're chasing rabbits down the wrong holes and claiming victory because you found an orange plastic cap that you're calling a carrot.
This isn't reducing the likelihood to succeed by 5%; if applied to a d20 it's essentially reducing it by 2.5%, so it's as though we made the d20 have a mean of 10, like the 2*3d6-11.

But that's not actually quite want we want --- I did mess that up in hastily firing off an idea that I hadn't had a chance to sit down and work through yet. What we want is to apply the confirmation die in the 3d6 case, not the d20 case. We either (1) roll 2*3d6-10, applying the confirmation mechanic, which should be comparable to an unaltered d20 system, or (2) just roll 3d6 with the confirmation mechanic, and compare that to a d20 system where bonuses are doubled together with DCs' distance from 10.

Let's look at (1) first. Here are the probabilities of meeting each target natural roll from 1 to 20, comparing vanilla 1d20 (no changes to DCs or modifiers), compared to 2*3d6-10 with the confirmation correction:

Once we correct for the mean of 2*3d6-10 being off by a half, the approximation error is symmetric around the DC/bonus combinations with a 50% success rate. The 3d6 curve is of course nonlinear -- no points for pointing that out; everyone already recognized that -- and so the discrepancy in success rates is not constant, but it is at worst 4%.
I don't know what you're actually graphing, here. Your d20 line has data outside of the possible range, and the value for 1 on d20 is incorrect. The rest are correct (2-20), but these obvious issues make me very leery of what it is you're doing.

Secondly, you've done exactly what I said the OP did -- you've tossed data from the scaled 3d6. Here, you've actually truncated the values form the graph which makes it look even more visually similar. This is bad math.

I think you've set yourself up to fail, here, because your next bit is way, way, way off the rails.
And here are the probabilities for a "reduced randomness" scheme: 3d6 with a confirmation correction and no change to DCs or modifiers, compared to 1d20 with doubled modifiers and new DCs set to 10 + 2*(DC - 10) -- that is to say, expanding around 10 by a factor of 2. (The x-axis is the target 3d6 roll; the roll needed on the d20 is different)

As advertised, using 3d6 produces less variation in rolls. But if we adjust modifers and DCs we achieve the same effect with a d20. We do have a bit of an issue on the extremes, in that the scaled d20 approximation makes some rolls guaranteed or impossible, whereas they are around 96% or 4% with 3d6. But again that's something everyone has acknowledged throughout this thread, so there's nothing special in you pointing that out -- it's not a problem with anyone's math, it's a fact that distributional approximations like this have the worst fit in the tails.
I have no idea what you've graphed here. The d20 in your scheme does not display a different CPDF -- it's still linear. But, here, you have a weird hybrid curve that goes to a 0% probability when you need a 16. Can you not even roll a 16 on a d20? Clearly, you can, you can roll all the way up to a 20 (not featured on your graph), so whatever you've done here, it's not the density function of a d20.

The best I can figure is that you've graphed a d20 scaled by 1/2 and recentered. You've increased the slope of the d20 line, but tossed a bunch of data. Your d20 values go from being able to generate a value of 5 through 16. This is NOT what a d20 can generate.

I also tried to understand your graph as using the expanded d20 roll as graphing likelihood of meeting a DC, but I can't get that math to work, either, even trying out a number of possible mistakes.

I was able to recreate your graph, though. I divided a d20's probability by 2 and recentered it at 10.5, then changed the first and last data points to be an average of the data point before and the data point after (which is why you have those ramps to 100% and 0% on the d20 line). I then recentered the 3d6 line to 10.5. I don't really know why we did this, because both lines were centered on 11, but whatever, it's a bunch of arbitrary decisions to make lines look like each other so it's all the same.

 roll d20 3d6 scaled d20 1​ 100​ 1.5​ 97.5​ 2​ 95​ 2.5​ 92.5​ 100​ 3​ 90​ 99.76852​ 3.5​ 87.5​ 99.53704​ 100​ 4​ 85​ 98.84259​ 100​ 4.5​ 82.5​ 98.14815​ 100​ 5​ 80​ 96.75926​ 100​ 5.5​ 77.5​ 95.37037​ 97.5​ 6​ 75​ 93.05556​ 95​ 6.5​ 72.5​ 90.74074​ 90​ 7​ 70​ 87.26852​ 85​ 7.5​ 67.5​ 83.7963​ 80​ 8​ 65​ 78.93519​ 75​ 8.5​ 62.5​ 74.07407​ 70​ 9​ 60​ 68.28704​ 65​ 9.5​ 57.5​ 62.5​ 60​ 10​ 55​ 56.25​ 55​ 10.5​ 52.5​ 50​ 50​ 11​ 50​ 43.75​ 45​ 11.5​ 47.5​ 37.5​ 40​ 12​ 45​ 31.71296​ 35​ 12.5​ 42.5​ 25.92593​ 30​ 13​ 40​ 21.06481​ 25​ 13.5​ 37.5​ 16.2037​ 20​ 14​ 35​ 12.73148​ 15​ 14.5​ 32.5​ 9.259259​ 10​ 15​ 30​ 6.944444​ 5​ 15.5​ 27.5​ 4.62963​ 2.5​ 16​ 25​ 3.240741​ 0​ 16.5​ 22.5​ 1.851852​ 0​ 17​ 20​ 1.157407​ 0​ 17.5​ 17.5​ 0.462963​ 18​ 15​ 18.5​ 12.5​ 19​ 10​ 19.5​ 7.5​ 20​ 5​

NOTE: I did some fast and dirty extrapolation for the 3d6 curve for the .5 values by averaging the preceding and following values. This is because there isn't a probability for rolling, say, 6.5 on 3d6 but it's a pain to get Excel to graph data where values are missing.
Note that in the latter case, the curves cross at 10.5, not at 11. This is due to the fact that by altering our DCs, tasks that previously had DC 11+modifier are no longer 50% success rate tasks; they've gotten a little more difficult, whereas the 50% success rate now sits between old DC 10+modifier (still DC 10+modifier) and DC 11+modifier (now DC 12+modifier). Introducing the confirmation mechanic on the 3d6 roll does the same thing: you no longer necessarily succeed on a DC 11+modifier task if you roll a natural 11; you only have a 50% chance.
Sigh. That makes things easier -- you need a 10.5 to succeed 50% of the time instead of a larger value, 11. This actually increases the likelihood of success by a small amount. That is, if what you've done was remotely correct at all. I point this out because you've not only managed to do horrible things to math, but you've failed to even apply a correct interpretation to the horrible things you've done as if they were okay.

Look, no matter what you do to target numbers, the CPDF of a d20 doesn't change. If you don't get a straight line from 100% to 5% from 1 to 20 with a slope of exactly -1, you've done something horribly wrong.

#### Ruin Explorer

##### Legend
Yes, but it pays to consider what you really want out of it. "Limit extremes" is a means, not an end in and of itself. What things actually happening in your game do you not want to happen?

Note that 5e already comes with bounded accuracy.

I concur. Changes like this should always be made with careful reflection on the purpose. Easily the biggest problem with house rules and modifications is that, especially pre-2000, most are/were made without considering the actual goal, instead either being "just because" or reactionary in design (reactionary here meaning "A thing I disliked happened so let's make a rule to prevent it!" without considering how it fits into the greater scheme of things).

I think if 5E used actual fumble rules beyond a 1 simply always missing, then that alone would go a long way towards justifying, say, 2d10 instead of 1d20, because such rules can easily turn a game into a farce. But it doesn't. Of course a lot of groups use very ill-considered "just because" type fumble rules, and I can certainly see that after adopting such, one might be tempted to switch to 2d10 or the like. I saw too many cases in 2E and 3E and other 90s and 00s games where people had whole chains of house rules that were there to compensate for the results of other house rules to think that's a good idea, though.

#### Coroc

##### Hero
I still simply do not get what should be achieved with the 3d6 instead of 1d20.
You got to count the faces of three dice, so you waste more time when doing rolls for fast actions.

You make situations which a good DM attaches a threshold to (Bend some bars, read some arcane runes etc.) less likely to occur but you do not reduce the probability of these to 0, which you would, if you just used a threshold!

You make skill checks more reliable, ok I give you that. But what consequences has this for your game?
If you want skilled characters to almost auto-succeed then just give them that on standard tasks and only require a roll on extraordinary displays of their skill.

For saves and attacks which rely on so much else I do not see a point also you destroy the fun of criticals (which in 5e have far less dire consequences than in other editions or systems)

#### Esker

##### Hero
Not what you say above, and still a really weird, non-mathematical thing to do to correct at a single point in the distribution because your choice of arbitrary centering (to get, recall, a optical match but not actual match in probabilities) to look better. You're chasing rabbits down the wrong holes and claiming victory because you found an orange plastic cap that you're calling a carrot.

I'm nearing the end of my patience with you, because you're so quick to fire insults and shoot down straw men without actually taking time to consider what anyone else is saying. You keep fixating on the fact that the probabilities don't actually match, and, no of course they don't match exactly. No one was claiming they were an exact match. But they're close, because we've matched the means and standard deviations of the die rolls.

I don't know what you're actually graphing, here. Your d20 line has data outside of the possible range, and the value for 1 on d20 is incorrect. The rest are correct (2-20), but these obvious issues make me very leery of what it is you're doing.

I've only graphed the even target rolls, since 2*3d6-10 can only produce even results before the confirmation correction. Obviously the probability of 1 or higher on a d20 is 1, but that's not on the graph; it's just interpolating between 0 and 2.

If the target is an odd value then the confirmation step doesn't matter because you'll either exceed the target or you'll fall below it, so you can use the probability of hitting one number higher without a confirmation correction. I didn't bother incorporating that because it doesn't change the shape, but since you're clearly intent on fixating on any small missing detail, here's the graph with the odd targets added in.

Secondly, you've done exactly what I said the OP did -- you've tossed data from the scaled 3d6. Here, you've actually truncated the values form the graph which makes it look even more visually similar. This is bad math.

I don't know what you mean by saying that I've tossed data. Unless you mean that I omitted odd-numbered targets. Well they're there now. Happy?

I have no idea what you've graphed here. The d20 in your scheme does not display a different CPDF -- it's still linear. But, here, you have a weird hybrid curve that goes to a 0% probability when you need a 16. Can you not even roll a 16 on a d20? Clearly, you can, you can roll all the way up to a 20 (not featured on your graph), so whatever you've done here, it's not the density function of a d20.

I explained this in my last post: the x-axis shows the target values for the 3d6 scheme. The corresponding targets on a d20 are different because we've changed the bonuses and DCs for that scheme only. If the 3d6 target is X, then the d20 target with rescaled DCs is 10 + 2*(X-10). So the points on the graph at 16 are the probability of getting 16 on 3d6 and the probability of getting 22 on a d20, since these are corresponding rolls. This is a linear transformation and so of course the CDF is still linear. Not sure why you seem to think it wouldn't be.

The best I can figure is that you've graphed a d20 scaled by 1/2 and recentered. You've increased the slope of the d20 line, but tossed a bunch of data. Your d20 values go from being able to generate a value of 5 through 16. This is NOT what a d20 can generate.

Yes, if you prefer you can think of the modified d20 scheme as altering the roll itself instead of the DC: roll the d20, add 10, and divide by 2, rounding down. So the maximum result is 15 and the minimum is 5 (but you only get 5 when the die shows a 1). And so, as I acknowledged, the adjusted d20 is missing a small chance of getting a 3-4, as well as a small chance of getting 16-18: with the confirmation correction, 3d6 has a 96.8% chance of rolling 5 or higher (instead of 100%), and a 3.2% chance of rolling 16 or higher (instead of 0%).

I point this out because you've not only managed to do horrible things to math, but you've failed to even apply a correct interpretation to the horrible things you've done as if they were okay.

I suggest taking a step back and reconsidering your stance that people who produce results that you don't follow must necessarily be mathematically inept, and instead begin from a place of good faith, reading what they have written and trying to make sense of it to the best of your ability before jumping on any perceived missing or incorrect detail as proof that the whole approach is "horrible math". I acknowledge having made some minor errors here and there, and I originally proposed the confirmation correction in a quick post from my phone with what turned out to be some details that were off before I had a chance to sit down and work out the numbers. But I promise you, my grasp of the underlying math is solid (as I believe I've demonstrated in our previous interactions here, so I figured you might extend some benefit of the doubt).

#### Esker

##### Hero
I still simply do not get what should be achieved with the 3d6 instead of 1d20.
You got to count the faces of three dice, so you waste more time when doing rolls for fast actions.

The OP's point was that you don't need to use 3d6 to achieve the goals that people have in mind when using 3d6 (namely, making bonuses matter more, and randomness matter less). You can just scale bonuses and adjust DCs, and stick with a d20 roll, to achieve nearly the same success rates that 3d6 gives you, but you've offloaded the mental effort to adjustments you can do away from the table, so that people don't have to add as many dice during the game.

#### NotAYakk

##### Legend
The OP's point was that you don't need to use 3d6 to achieve the goals that people have in mind when using 3d6 (namely, making bonuses matter more, and randomness matter less). You can just scale bonuses and adjust DCs, and stick with a d20 roll, to achieve nearly the same success rates that 3d6 gives you, but you've offloaded the mental effort to adjustments you can do away from the table, so that people don't have to add as many dice during the game.
Well, rather that when people describe why they want to use 3d6, they actually don't describe that. Often they talk about liking the "bell curve".

Almost all of the effect of 3d6 using is equivalent to a re-scaling of bonuses/DCs and using a d20. With the addendum that things in the top 5%/bottom 5% (corresponding to natural 20s and natural 1s, which often is played as auto-hit auto-miss even when D&D doesn't say so) aren't as well captured.

If people where rolling 3d6 with the intention of "making modifiers twice as important" and viewed the cost of adding up 3 dice each roll as being acceptable for that, more power to them. But when I people talk about 3d6 that, in my experience isn't how they frame it. Which leads me to suspect they actually think it is doing something else other than what it does.

Now, sometimes slowing down gameplay with "busywork" is worthwhile; D&D with combat reduced to a single weighted coin flip wouldn't be as fun. Rolling 3d6 for skill checks, when skill checks tend to be high impact, might be worth it for that reason.

But I still hold my position that if you want to do something like that, you should make it more interesting; a minigame. You roll 3d6 for a skill check. If the result is under the DC, the DM produces a malus (a cost), and you can say what else you are doing to try to succeed. When you do that, you get to pick up once of your 3d6 and reroll it.

So if you are sneaking into a castle, you roll 3d6+9 against a 20 DC. You roll a 1, 3, 5 and get a total of 18. The DM says as you cross a courtyard, a guard yells out "who goes there" (malus: castle alert level just went up).

You use your mimicry ability to make the sound of a cat yowl. The DM lets you reroll your 1 -- you get a 3. 3+3+5+9=20, so now you have successfully snuck into the castle. Except some of the guards are on a bit higher alert, so after you convince the princess to leave with you it will be a bit harder, which could lead to you having a 10 minute head start instead of a 4 hour one.

But that isn't central to my thesis here.

#### Ovinomancer

##### No flips for you!
I'm nearing the end of my patience with you, because you're so quick to fire insults and shoot down straw men without actually taking time to consider what anyone else is saying. You keep fixating on the fact that the probabilities don't actually match, and, no of course they don't match exactly. No one was claiming they were an exact match. But they're close, because we've matched the means and standard deviations of the die rolls.
I haven't insulted you, I've been emphatic that what you're doing is wrong, mathematically.

I've only graphed the even target rolls, since 2*3d6-10 can only produce even results before the confirmation correction. Obviously the probability of 1 or higher on a d20 is 1, but that's not on the graph; it's just interpolating between 0 and 2.

If the target is an odd value then the confirmation step doesn't matter because you'll either exceed the target or you'll fall below it, so you can use the probability of hitting one number higher without a confirmation correction. I didn't bother incorporating that because it doesn't change the shape, but since you're clearly intent on fixating on any small missing detail, here's the graph with the odd targets added in.

I don't know what you mean by saying that I've tossed data. Unless you mean that I omitted odd-numbered targets. Well they're there now. Happy?

Here's the full graph for 2*3d6-10 against d20:

Perhaps you might not the missing data now?

I explained this in my last post: the x-axis shows the target values for the 3d6 scheme. The corresponding targets on a d20 are different because we've changed the bonuses and DCs for that scheme only. If the 3d6 target is X, then the d20 target with rescaled DCs is 10 + 2*(X-10). So the points on the graph at 16 are the probability of getting 16 on 3d6 and the probability of getting 22 on a d20, since these are corresponding rolls. This is a linear transformation and so of course the CDF is still linear. Not sure why you seem to think it wouldn't be.
Wait, you presented data on two different x-axis scales with a single label? This didn't ring any alarm bells for you? Your graph shows that the odds of rolling a 16 on 3d6 is similar to the odds of rolling a 22 on d20 by putting both points on the 16?

Also, how, exactly, do you roll a 22 on d20? That's a 0 percent chance. How did that not ring an alarm bell for you?

Yes, if you prefer you can think of the modified d20 scheme as altering the roll itself instead of the DC: roll the d20, add 10, and divide by 2, rounding down. So the maximum result is 15 and the minimum is 5 (but you only get 5 when the die shows a 1). And so, as I acknowledged, the adjusted d20 is missing a small chance of getting a 3-4, as well as a small chance of getting 16-18: with the confirmation correction, 3d6 has a 96.8% chance of rolling 5 or higher (instead of 100%), and a 3.2% chance of rolling 16 or higher (instead of 0%).
This doesn't change the odds of rolling any single value, though. Again, you've just reshaped the distribution and then pretended it's the same thing -- it's not the same thing. You've just acknowledged that you're limiting the d20 roll to between 5 and 15 in half step increments against the same DC as the 3d6 to get a partial match in shape. But, AGAIN, the lines looking the similar with your do mathemagic doesn't actually make them the same thing -- you've lost any ability to compare and are just fooling yourself.

The OP method doesn't do what's advertised, it, instead, limits the effective range of the d20 versus the new DC schema and pretends the limited scope means a d20 roll isn't much different from a 3d6 roll. It's bunk mathematically and bunk logically.

I suggest taking a step back and reconsidering your stance that people who produce results that you don't follow must necessarily be mathematically inept, and instead begin from a place of good faith, reading what they have written and trying to make sense of it to the best of your ability before jumping on any perceived missing or incorrect detail as proof that the whole approach is "horrible math". I acknowledge having made some minor errors here and there, and I originally proposed the confirmation correction in a quick post from my phone with what turned out to be some details that were off before I had a chance to sit down and work out the numbers. But I promise you, my grasp of the underlying math is solid (as I believe I've demonstrated in our previous interactions here, so I figured you might extend some benefit of the doubt).
I recreated your graph exactly above, to the extend that you agreed that my assessment of the d20 slope is TRUE. How you can claim I don't understand your points is beyond me -- I very much grasp what you're putting down. It's just WRONG. I've tried multiple times to show that what you're doing is fooling yourself with an improper analysis -- it's you that isn't following.

#### Ovinomancer

##### No flips for you!
Well, rather that when people describe why they want to use 3d6, they actually don't describe that. Often they talk about liking the "bell curve".

Almost all of the effect of 3d6 using is equivalent to a re-scaling of bonuses/DCs and using a d20. With the addendum that things in the top 5%/bottom 5% (corresponding to natural 20s and natural 1s, which often is played as auto-hit auto-miss even when D&D doesn't say so) aren't as well captured.

If people where rolling 3d6 with the intention of "making modifiers twice as important" and viewed the cost of adding up 3 dice each roll as being acceptable for that, more power to them. But when I people talk about 3d6 that, in my experience isn't how they frame it. Which leads me to suspect they actually think it is doing something else other than what it does.

Now, sometimes slowing down gameplay with "busywork" is worthwhile; D&D with combat reduced to a single weighted coin flip wouldn't be as fun. Rolling 3d6 for skill checks, when skill checks tend to be high impact, might be worth it for that reason.

But I still hold my position that if you want to do something like that, you should make it more interesting; a minigame. You roll 3d6 for a skill check. If the result is under the DC, the DM produces a malus (a cost), and you can say what else you are doing to try to succeed. When you do that, you get to pick up once of your 3d6 and reroll it.

So if you are sneaking into a castle, you roll 3d6+9 against a 20 DC. You roll a 1, 3, 5 and get a total of 18. The DM says as you cross a courtyard, a guard yells out "who goes there" (malus: castle alert level just went up).

You use your mimicry ability to make the sound of a cat yowl. The DM lets you reroll your 1 -- you get a 3. 3+3+5+9=20, so now you have successfully snuck into the castle. Except some of the guards are on a bit higher alert, so after you convince the princess to leave with you it will be a bit harder, which could lead to you having a 10 minute head start instead of a 4 hour one.

But that isn't central to my thesis here.
Rolling 3d6 doesn't make modifiers twice as important, though. I get that's your theory, which you present by truncating and scaling the d20 roll by a factor of 1/2. This loses data to the fractional results. I get that you modify how ability modifiers are used to take a bit of advantage of those fractional results (and die bonuses as well), but that's only a small selection of modifiers, most of which won't take advantage of that. Any system that says that you're going to truncate and then toss data to replicate a different system is questionable, to start with.

What 3d6 does is make the ends much less common and focuses much of the likely results in the middle. It also alters the assumptions of success to improving likelihood for lower than average needed rolls and decreasing it for the over average needed rolls. This is not at all like your schema, which only approximates the middle 2/3rds of the 3d6 roll by tossing (roughly) half of your data.

#### Esker

##### Hero
EDIT: Fixed

@Ovinomancer, for the sake of transparency, here are some tables showing the various quantities that go into my graphs, so you can more easily check my results against AnyDice or whatever. I hadn't shown this before because I was doing my calculations in R code rather than in a spreadsheet, but I've added the intermediate columns for you.

Here are the success rates for the ordinary 1d20 compared to 2*3d6-10, where the latter uses a confirmation correction.

Code:
``````   Target P_success_1d20 ScaledTarget P_exact_3d6 P_at_or_above_3d6 P_success_scaled3d6
1       0           1.00          5.0       0.028             0.981               0.968
2       1           1.00          5.5       0.000             0.954               0.954
3       2           0.95          6.0       0.046             0.954               0.931
4       3           0.90          6.5       0.000             0.907               0.907
5       4           0.85          7.0       0.069             0.907               0.873
6       5           0.80          7.5       0.000             0.838               0.838
7       6           0.75          8.0       0.097             0.838               0.789
8       7           0.70          8.5       0.000             0.741               0.741
9       8           0.65          9.0       0.116             0.741               0.683
10      9           0.60          9.5       0.000             0.625               0.625
11     10           0.55         10.0       0.125             0.625               0.562
12     11           0.50         10.5       0.000             0.500               0.500
13     12           0.45         11.0       0.125             0.500               0.438
14     13           0.40         11.5       0.000             0.375               0.375
15     14           0.35         12.0       0.116             0.375               0.317
16     15           0.30         12.5       0.000             0.259               0.259
17     16           0.25         13.0       0.097             0.259               0.211
18     17           0.20         13.5       0.000             0.162               0.162
19     18           0.15         14.0       0.069             0.162               0.127
20     19           0.10         14.5       0.000             0.093               0.093
21     20           0.05         15.0       0.046             0.093               0.069
22     21           0.00         15.5       0.000             0.046               0.046``````

The way the final column is calculated is by applying the confirmation die, which subtracts half the probability of meeting the target exactly from the overall success rate.

And here's the table for the ordinary 3d6 (with a confirmation correction), compared to a rescaled 1d20 (which you can either think of as transforming the target value to 10 + 2*(Target-10), or as transforming the roll itself to 10+(Roll-10)/2).

Code:
``````   Target P_exact_3d6 P_at_or_above_3d6 P_success_3d6 ScaledTarget P_success_scaled1d20
1       2       0.000             1.000         1.000           -6                 1.00
2       3       0.005             1.000         0.998           -4                 1.00
3       4       0.014             0.995         0.988           -2                 1.00
4       5       0.028             0.981         0.968            0                 1.00
5       6       0.046             0.954         0.931            2                 0.95
6       7       0.069             0.907         0.873            4                 0.85
7       8       0.097             0.838         0.789            6                 0.75
8       9       0.116             0.741         0.683            8                 0.65
9      10       0.125             0.625         0.562           10                 0.55
10     11       0.125             0.500         0.438           12                 0.45
11     12       0.116             0.375         0.317           14                 0.35
12     13       0.097             0.259         0.211           16                 0.25
13     14       0.069             0.162         0.127           18                 0.15
14     15       0.046             0.093         0.069           20                 0.05
15     16       0.028             0.046         0.032           22                 0.00
16     17       0.014             0.019         0.012           24                 0.00
17     18       0.005             0.005         0.002           26                 0.00
18     19       0.000             0.000         0.000           28                 0.00``````

Last edited:

#### Esker

##### Hero
I've been emphatic that what you're doing is wrong, mathematically.

And yet, you have yet to give any justification for that claim, other than repeatedly harping on the fact (a fact that everyone acknowledges) that the range of 3d6 is not identical to the range of 1d20, and so for extreme DCs in either direction, you have one method giving a small chance of either success or failure and the other giving zero chance. That may or may not be a problem in a game. It's not a mathematical flaw, but a consequence of a particular choice of approximation.

Perhaps you might not the missing data now?

There's no missing data, it's just that the d20 probabilities hit the ceiling and the floor. Again, you wind up with one method giving you probabilities very near 0 or 1, and the other giving you probabilities exactly at zero or 1.

Wait, you presented data on two different x-axis scales with a single label? This didn't ring any alarm bells for you?

I did this because that correspondence is the claim. On the one hand, you have a 3d6 roll, with one set of modifiers and DCs, and then you have a translation of those modifiers and DCs to a d20 system. The target value changes when you change the modifiers and DCs, but there's a direct correspondence between one roll and the other. That was the original claim

Your graph shows that the odds of rolling a 16 on 3d6 is similar to the odds of rolling a 22 on d20 by putting both points on the 16?

Yes, because 16 maps to 22 when you work out the correspondence implied by the modified bonuses and DCs. If you needed a natural 16 to hit some AC originally, then after you double attack bonuses and move ACs twice as far away from 10, you're going to need a natural 22 to hit. For example, if your attack bonus was +5, then needing a natural 16 means you're up against an AC of 21. That AC becomes 32, and your bonus becomes +10. Hence, you need a natural 22. I'll leave other combinations as an exercise for the reader, but they're all the same: any time you needed a 16 with the original modifiers and DCs, you're going to need a 22 now.

Also, how, exactly, do you roll a 22 on d20? That's a 0 percent chance. How did that not ring an alarm bell for you?

You don't. That's why there's a 0% success chance if the target value is a 22. I'm not sure what's so hard about that.

You've just acknowledged that you're limiting the d20 roll to between 5 and 15 in half step increments against the same DC as the 3d6 to get a partial match in shape.

I didn't limit the d20 roll to between 5 and 15. The 5 to 15 range is the target 3d6 roll. If instead of transforming the 3d6 to 2*3d6-10 you leave the 3d6 alone and transform the d20 using the inverse transformation, which is 10+(1d20-10)/2, then the range of possibilities is 5 to 15, where you get a 5 if you roll a natural 1, and a 15 if you roll a natural 20.

But, AGAIN, the lines looking the similar with your do mathemagic doesn't actually make them the same thing

No one is saying they are the same thing. We are saying they give very similar success probabilities.

The OP method doesn't do what's advertised, it, instead, limits the effective range of the d20 versus the new DC schema and pretends the limited scope means a d20 roll isn't much different from a 3d6 roll.

I think what you're not getting is that the actual range of possible rolls doesn't matter in and of itself, since we're always comparing the roll to a target. The only thing that affects outcomes is the probabilities of success at different combinations of bonuses and DCs. And we can simplify that further if instead of treating bonuses as something we add to our roll, we treat them as something that reduces the DC; then, the only thing that matters is probabilities of success vs different DCs.

How you can claim I don't understand your points is beyond me -- I very much grasp what you're putting down. It's just WRONG. I've tried multiple times to show that what you're doing is fooling yourself with an improper analysis -- it's you that isn't following.

You have repeatedly asserted that what I'm doing is wrong, but, apart from some early off-by-one type errors which I have corrected, you haven't actually pointed out any flaws. As far as I can tell, you just don't like the fact that some small probabilities are approximated by zero, and some large ones are approximated by 1. And that's fine as a critique; but don't pretend it reflects a mathematical mistake... it's a property that everyone has acknowledged from the very start.

#### Ovinomancer

##### No flips for you!
EDIT: Fixed
Your edit doesn't really help as you have some columns without labels at all (like the first 2) and your C&P doesn't make it clear which header goes with which column. I figured it out, though, as I wasn't actually confused as to how you got your data (and got the same data already).

For others, header followed by column of data:
Target -- column 2
P_Success_1d20 -- column 3
ScaledTarget -- column 4
P_exact_3d6 -- column 5
P_at_or_above_3d6 -- column 6
P_success_scaled_3d6 -- column 7

@Ovinomancer, for the sake of transparency, here are some tables showing the various quantities that go into my graphs, so you can more easily check my results against AnyDice or whatever. I hadn't shown this before because I was doing my calculations in R code rather than in a spreadsheet, but I've added the intermediate columns for you.

Here are the success rates for the ordinary 1d20 compared to 2*3d6-10, where the latter uses a confirmation correction.

Code:
``````   Target P_success_1d20 ScaledTarget P_exact_3d6 P_at_or_above_3d6 P_success_scaled3d6
1       0           1.00          5.0       0.028             0.981               0.968
2       1           1.00          5.5       0.000             0.954               0.954
3       2           0.95          6.0       0.046             0.954               0.931
4       3           0.90          6.5       0.000             0.907               0.907
5       4           0.85          7.0       0.069             0.907               0.873
6       5           0.80          7.5       0.000             0.838               0.838
7       6           0.75          8.0       0.097             0.838               0.789
8       7           0.70          8.5       0.000             0.741               0.741
9       8           0.65          9.0       0.116             0.741               0.683
10      9           0.60          9.5       0.000             0.625               0.625
11     10           0.55         10.0       0.125             0.625               0.562
12     11           0.50         10.5       0.000             0.500               0.500
13     12           0.45         11.0       0.125             0.500               0.438
14     13           0.40         11.5       0.000             0.375               0.375
15     14           0.35         12.0       0.116             0.375               0.317
16     15           0.30         12.5       0.000             0.259               0.259
17     16           0.25         13.0       0.097             0.259               0.211
18     17           0.20         13.5       0.000             0.162               0.162
19     18           0.15         14.0       0.069             0.162               0.127
20     19           0.10         14.5       0.000             0.093               0.093
21     20           0.05         15.0       0.046             0.093               0.069
22     21           0.00         15.5       0.000             0.046               0.046``````

The way the final column is calculated is by applying the confirmation die, which subtracts half the probability of meeting the target exactly from the overall success rate.
And, the arithmetic is correct, here. Everyone -- the above numbers are corrected calculated. But, that was never the problem.

The problem is that the 2*3d6-10 has a range of [-4,26] and increments in steps of 2. The d20 has a range of [1,20] and increments in steps of 1. You're actually only comparing the data here at 2, 4, 6, ..., 18, and 20. You've tossed half of the possible rolls of the d20 to compare against 2/3rds of the possible rolls of 3d6. When you do this, you not that halving a d20 roll looks a lot like the middle 2/3rds of a 3d6 roll (recentered) between the values of 6 and 16. Does that not make you stop and wonder what you actually did? Because, while I can say that 1+1 times 2 has the same result as 2+2, 1+1 is NOT like 2+2, even if my arithmetic was right.

And here's the table for the ordinary 3d6 (with a confirmation correction), compared to a rescaled 1d20 (which you can either think of as transforming the target value to 10 + 2*(Target-10), or as transforming the roll itself to 10+(Roll-10)/2).

Code:
``````   Target P_exact_3d6 P_at_or_above_3d6 P_success_3d6 ScaledTarget P_success_scaled1d20
1       2       0.000             1.000         1.000           -6                 1.00
2       3       0.005             1.000         0.998           -4                 1.00
3       4       0.014             0.995         0.988           -2                 1.00
4       5       0.028             0.981         0.968            0                 1.00
5       6       0.046             0.954         0.931            2                 0.95
6       7       0.069             0.907         0.873            4                 0.85
7       8       0.097             0.838         0.789            6                 0.75
8       9       0.116             0.741         0.683            8                 0.65
9      10       0.125             0.625         0.562           10                 0.55
10     11       0.125             0.500         0.438           12                 0.45
11     12       0.116             0.375         0.317           14                 0.35
12     13       0.097             0.259         0.211           16                 0.25
13     14       0.069             0.162         0.127           18                 0.15
14     15       0.046             0.093         0.069           20                 0.05
15     16       0.028             0.046         0.032           22                 0.00
16     17       0.014             0.019         0.012           24                 0.00
17     18       0.005             0.005         0.002           26                 0.00
18     19       0.000             0.000         0.000           28                 0.00``````
Why do you have negative numbers for the targets of the d20? Why are you comparing impossible results to possible results? This should send up warning flags, but it hasn't, yet. If you do the roll transformation, you don't see these, but you do see that what's you've done here is throw away half the of the d20 results, again. And truncate the data, again.

Here's the graph of the above using the roll adjust method (DC adjust just makes no sense). You can see you're only considering half of the d20 rolls in comparison to the 3d6 and you've tossed 6 data points off the 3d6 to do so.

These comparisons are bad math as you're not doing the same things to both sides of the equations and then claiming the results are similar.

#### Esker

##### Hero
Your edit doesn't really help as you have some columns without labels at all (like the first 2) and your C&P doesn't make it clear which header goes with which column.

The column headings are aligned in my browser, but apologies if they aren't in yours. Likely a difference in browser/font settings. Glad you were able to sort it out.

The problem is that the 2*3d6-10 has a range of [-4,26] and increments in steps of 2. The d20 has a range of [1,20] and increments in steps of 1. You're actually only comparing the data here at 2, 4, 6, ..., 18, and 20.

You are simply not listening. Yes, we all realize that 2*3d6-10 has a wider range than 1d20 does. I've accounted for the increment size through the confirmation die: if you meet the DC exactly you have to confirm with a 1d2 roll. If the target is even, this can happen, and you have a 50% chance of confirming. If the target is odd, this can't happen. So there are results for every target, that are different. Not just even ones.

When you do this, you not that halving a d20 roll looks a lot like the middle 2/3rds of a 3d6 roll (recentered) between the values of 6 and 16. Does that not make you stop and wonder what you actually did?

I am perfectly aware of what I did. A 3d6 roll nearly always produces results between 6 and 16. And you can approximate the probabilities by using a transformed 1d20 roll (or, equivalently, doubling bonuses and moving DCs twice as far from 10). This results in always getting results between 6 and 16, whereas 3d6 merely almost always falls in this range. Again, everyone has been up front about this.

Why do you have negative numbers for the targets of the d20? Why are you comparing impossible results to possible results?

Again, the targets are the values you need to meet or exceed to achieve a "success". It's perfectly well defined to say you need "at least -2" on the die to succeed in a check.

These comparisons are bad math as you're not doing the same things to both sides of the equations and then claiming the results are similar.

I am doing the corresponding things to both sides, as indicated by the (approximate) equivalence that was claimed.

Last edited:

#### Ovinomancer

##### No flips for you!
And yet, you have yet to give any justification for that claim, other than repeatedly harping on the fact (a fact that everyone acknowledges) that the range of 3d6 is not identical to the range of 1d20, and so for extreme DCs in either direction, you have one method giving a small chance of either success or failure and the other giving zero chance. That may or may not be a problem in a game. It's not a mathematical flaw, but a consequence of a particular choice of approximation.
I haven't given any justifications?! Well, I guess repeatedly explaining exactly how you've taken two things and compared a narrow range after arbitrarily adjusting one of the data sets (including changing scale, center, and tossing out inconvenient data points) could be construed as not providing justification. I mean, if you're going to immediately discount those points to get to the claim I haven't justified anything, at least.

I sense a pattern -- you toss data points to make your comparison and you toss data points to claim I don't have any.

There's no missing data, it's just that the d20 probabilities hit the ceiling and the floor. Again, you wind up with one method giving you probabilities very near 0 or 1, and the other giving you probabilities exactly at zero or 1.

The 3d6 probabilities do not, yet you've truncated them and ignored them to assume that the part you kept is similar.

I did this because that correspondence is the claim. On the one hand, you have a 3d6 roll, with one set of modifiers and DCs, and then you have a translation of those modifiers and DCs to a d20 system. The target value changes when you change the modifiers and DCs, but there's a direct correspondence between one roll and the other. That was the original claim
And that claim is incorrect. It is wrong. For the reasons I keep reiterating: you've tossed data and rescaled data and recentered data. What you've done is like zooming in on a small part of the arc of a circle and note that, zoomed it, it corresponds to a line. But the whole is not the part.

Yes, because 16 maps to 22 when you work out the correspondence implied by the modified bonuses and DCs. If you needed a natural 16 to hit some AC originally, then after you double attack bonuses and move ACs twice as far away from 10, you're going to need a natural 22 to hit. For example, if your attack bonus was +5, then needing a natural 16 means you're up against an AC of 21. That AC becomes 32, and your bonus becomes +10. Hence, you need a natural 22. I'll leave other combinations as an exercise for the reader, but they're all the same: any time you needed a 16 with the original modifiers and DCs, you're going to need a 22 now.

22 DOES NOT EXIST! There is NO mapping 22 on a d20 to ANYTHING!!!

When you model physical things, your constraints are the same as the physical thing. That you can do math on non=physical things doesn't make them suddenly exist.

You don't
. That's why there's a 0% success chance if the target value is a 22. I'm not sure what's so hard about that.

And yet, you've compared a 0% probability for getting a 22 on d20 to a non-zero probability of getting a 16 on 3d6. You've tossed data and made comparison end-points nonsensical.

I didn't limit the d20 roll to between 5 and 15. The 5 to 15 range is the target 3d6 roll. If instead of transforming the 3d6 to 2*3d6-10 you leave the 3d6 alone and transform the d20 using the inverse transformation, which is 10+(1d20-10)/2, then the range of possibilities is 5 to 15, where you get a 5 if you roll a natural 1, and a 15 if you roll a natural 20.

I am flabbergasted by this argument. You haven't limited the d20 roll to 5 and 15, you just get a 5 if you roll a 1 and a 15 if you roll a 20! All so very clear, now.

I mean, I make this argument to try to show that you're arbitrarily constraining data to a narrower range and not accounting for this fact in your conclusion, and you tell me that no, I don't understand, what you've done is arbitrarily constrain the data into a narrow range and, look, when that's ignored you get your conclusion!

No one is saying they are the same thing
. We are saying they give very similar success probabilities.
Then claim is that a scaled version of a d20 is the very similar to 3d6. To do this, you half the d20 roll and recenter it (this is, mathematically, the same, as you've acknowledged) so that it only covers part of the 3d6 roll, and note that, when this happens, the lines look similar. What you can't seem to grasp is that you've tossed at least a 1/3 of the data points to do this, meaning scaled d20 isn't similar to 3d6, it's similar to the range of 3d6 outcomes between 5 and 15 ONLY. And then only when you discard all of the fractional results of a halved d20 (there are no fractional results on 3d6, so I cannot compared to non-existent data points).

So, sure, if you scale a d20, toss half of those results, and compare to the center part of 3d6, you get some similarity. Yay? This is totally not saying that scaled d20 corresponds to 3d6, though.

I think what you're not getting is that the actual range of possible rolls doesn't matter in and of itself, since we're always comparing the roll to a target. The only thing that affects outcomes is the probabilities of success at different combinations of bonuses and DCs. And we can simplify that further if instead of treating bonuses as something we add to our roll, we treat them as something that reduces the DC; then, the only thing that matters is probabilities of success vs different DCs.
The... actual range of possible rolls... doesn't matter when comparing actual rolls? :headdesk:

This is why I said mathturbation early in the thread. This is ignoring the physical limits of the system being modeled and claiming that the model of unphysical results is true. This is the greatest sin in using statistics -- that of reification or mistaking the model for reality.

You have repeatedly asserted that what I'm doing is wrong, but, apart from some early off-by-one type errors which I have corrected, you haven't actually pointed out any flaws. As far as I can tell, you just don't like the fact that some small probabilities are approximated by zero, and some large ones are approximated by 1. And that's fine as a critique; but don't pretend it reflects a mathematical mistake... it's a property that everyone has acknowledged from the very start.
Your assumptions are wrong, which makes the statistics wrong. You've swapped an unphysical model for reality and claimed victory because the math worked out. ALWAYS examine your assumptions when doing statistical analysis. You can run a model on just about anything and it will give you an answer, but it is often not the right answer. Your error is in conception, which makes your math wrong before you even start.

#### Ovinomancer

##### No flips for you!
The column headings are aligned in my browser, but apologies if they aren't in yours. Likely a difference in browser/font settings. Glad you were able to sort it out.

You are simply not listening. Yes, we all realize that 2*3d6-10 has a wider range than 1d20 does. I've accounted for the increment size through the confirmation die: if you meet the DC exactly you have to confirm with a 1d2 roll. If the target is even, this can happen, and you have a 50% chance of confirming. If the target is odd, this can't happen. So there are results for every target, that are different. Not just even ones.
You're failing to understand. If I roll a 1 on a d20, there is no existing roll on 2*3d6-10 that is also a 1. I cannot compare physical, real events that do not exist. Extrapolation doesn't change this.

I am perfectly aware of what I did. A 3d6 roll nearly always produces results between 6 and 16. And you can approximate the probabilities by using a transformed 1d20 roll (or, equivalently, doubling bonuses and moving DCs twice as far from 10). This results in always getting results between 6 and 16, whereas 3d6 merely almost always falls in this range. Again, everyone has been up front about this.
We're down to "nearly always" being about 90% of the time, now? Ask your players if they're okay removing 1 and 20 from the game.

Again, the targets are the values you need to meet or exceed to achieve a "success". It's perfectly well defined to say you need "at least -2" on the die to succeed in a check.

Unphysical results in models of physical events are NEVER well defined.
I am doing the corresponding things to both sides, as indicated by the (approximate) equivalence that was claimed.
You are most assuredly NOT doing the same thing to both sides. That's the point, it's how you got here to begin with.

#### Esker

##### Hero
The 3d6 probabilities do not, yet you've truncated them and ignored them to assume that the part you kept is similar.

Ok, since you continue to be hung up on the fact that my graph ends before the 3d6 curve gets to the top and bottom, here:

And for good measure, here's a graph of the differences in success probabilities at each adjusted DC (think of the x-axis of all of these graphs as the DC of the check minus the modifier).

So, across the range of adjusted DCs, the two methods yield success probabilities within 4.5% of each other; essentially, depending on the DC, switching from one to the other will give some characters the equivalent of somewhere between a -1 and +1.

For the reasons I keep reiterating: you've tossed data and rescaled data and recentered data.

Now hopefully we can agree that I haven't tossed any data, as I'm showing the full range of possibilities.

22 DOES NOT EXIST! There is NO mapping 22 on a d20 to ANYTHING!!!

It exists as a target, not as a possible roll. If you have a 25 AC and are facing a monster with a +3 to hit, then the adjusted DC of their roll is 22. RAW they auto-hit you on a 20, but omitting that, they need a 22 to hit you, which means they can't.

And yet, you've compared a 0% probability for getting a 22 on d20 to a non-zero probability of getting a 16 on 3d6.

Yes. I'm comparing 0% to roughly 3% and saying those are close. You can object for gameplay reasons (essentially, using this approximation makes some tasks that would be very easy instead automatic, and some that should be very difficult impossible; or vice versa, depending on which you're treating as the reference method and which the approximation method). But it's not a mathematical error.

I am flabbergasted by this argument. You haven't limited the d20 roll to 5 and 15, you just get a 5 if you roll a 1 and a 15 if you roll a 20! All so very clear, now.

Yes, if you roll a 1, and apply 10 + (roll-10)/2 (rounding down when halving), you get 5. And if you roll a 20, and apply 10 + (roll-10)/2, you get 15. This isn't some purely theoretical exercise. You could, in principle, do that math with your rolls at the table. That's not what @NotAYakk was actually suggesting, but adjusting the die rolls like that is mathematically equivalent to doubling your bonus and doubling the DCs' distance from 10.

when that's ignored you get your conclusion!

I haven't ignored anything. I've been entirely up front all along (as was the OP) about what happens with extreme DCs. The approximation is still good at those extremes as measured by differences in probability. You might not consider approximating 3% with 0% or vice versa to be a good approximation, and that's fine. That's a matter of gaming priorities, not math.

What you can't seem to grasp is that you've tossed at least a 1/3 of the data points to do this, meaning scaled d20 isn't similar to 3d6, it's similar to the range of 3d6 outcomes between 5 and 15 ONLY.

You keep saying this but I haven't tossed out anything.

The... actual range of possible rolls... doesn't matter when comparing actual rolls? :headdesk:

Right, because nobody is saying that 3d6 produces similar rolls to rescaled d20 (or vice versa). We are saying that if you use a suitable rescaling that (approximately) equalizes the variance of the two distributions, then the success probabilities are close, for any DC you want to set.

You've swapped an unphysical model for reality and claimed victory because the math worked out.

There's nothing unphysical about any of this. It's all something you could do in your game. Either (1) roll 3d6 to resolve checks, double the result, and subtract 10. If the result ties the DC, confirm success with a d2; or (2) roll 1d20 to resolve checks, as written. The claim is that these produce very similar success probabilities, regardless of the DC.

Alternatively, if you want luck to play less of a role in your game, you can either (1) roll 3d6 to resolve checks, confirming ties with a d2; or (2) roll 1d20, halve the distance from 10 and then add 10; or (3) double all bonuses and stretch DCs to be DC' = 10 + 2*(DC - 10). (2) and (3) are exactly identical; (1) is very close, at all DCs.

Any of these are things you could actually do; they're not impractical thought experiments.

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#### Esker

##### Hero
You're failing to understand. If I roll a 1 on a d20, there is no existing roll on 2*3d6-10 that is also a 1. I cannot compare physical, real events that do not exist. Extrapolation doesn't change this.

There is if you use the confirmation die. Rolling a 6 on the 3d6 yields a 2 after the transformation. You then have a 50% chance of that turning into a 1. Outside of critical fumbles or whatever, the distinction between rolling a 1 and rolling a 2 only matters if the number you need to succeed (i.e., the adjusted DC) is a 2, so there's no point in rolling to confirm unless you wind up with exactly 6 on the dice.

We're down to "nearly always" being about 90% of the time, now? Ask your players if they're okay removing 1 and 20 from the game.

With the confirmation mechanic it's about 93.5%. But it's not actually sensible to add the two discrepancies together, since (again, setting aside special outcomes on 1s and 20s) they're never relevant at the same time. There might be some DCs where the 3d6 method has a 3.2% chance of succeeding whereas the d20 method has 0. But those are not the same DCs when the 3d6 method has a 3.2% chance of failing and the d20 method is guaranteed to succeed.

Unphysical results in models of physical events are NEVER well defined.

You keep saying this but I don't know what you mean by it. We're talking about rolling dice and doing math on the results. It's as physical as anything else in D&D, even if some of the calculations wind up being slightly more complicated.

You are most assuredly NOT doing the same thing to both sides. That's the point, it's how you got here to begin with.

No, not the same thing, but the analogous thing, according to the correspondence that was proposed. The original claim (to refresh your memory) was not that 3d6 and 1d20 are the same. The claim was that 3d6 is approximately equivalent to 1d20 if you adjust bonuses and DCs (but only for the d20). So you're not comparing a roll in one method to the same roll in the other; you're comparing a success probability in one system using the original bonuses and DCs, to the success probability of the same check using the other roll method after applying the necessary adjustments to bonuses and DC in the latter case only.

It's a bit like comparing monetary values across time: if you wanted to say that "the median U.S. household has similar buying power today as they did 30 years ago", it would be flat out wrong to compare median 2020 income with median 1990 household income directly. You have to multiply one side (but not the other) by an inflation adjuster to be able to make any sense of the comparison, because there's a correspondence between \$ in 2020 and \$ in 1990. In my graphs I've done the equivalent of adjusting the 1990 dollar values for inflation so that they are in 2020 terms, and left the 2020 values alone.

Or, perhaps an even more apt comparison: suppose you wanted to claim that "Classroom A and Classroom B have similar academic aptitude, as measured by standardized tests." But one class all took the SAT and the other took the ACT. You can't compare their scores directly; you have to recenter and rescale to make any kind of meaningful comparison. You could do that either by converting both to z-scores, say, or you could convert ACT scores to SAT equivalents based on their z-scores.

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