Proficiency Die, and why stop there?

[Kane0]

Had a late night thought about this seemingly uncommon DMG optional rule. I'm guessing (stabbing in the dark really) that people don't like it because it increases random swing of a roll in a situation where the d20 already swings plenty enough already, but what if when you used the proficiency die you also applied the same concept to your attribute bonus? So say a +1 modifier from strength would correspond with a +1d4, up to a +5 giving you a +1d12 on top of the proficiency die (+6 through to +10 giving another die again?)
This way the coupling of the proficiency die and the attribute die to the d20 would make for a stronger bell curve effect, especially when you throw in things like advantage, expertise and effects like Bless. The averages aren't exact either, so that's probably going to get the statisticians excited for a bit.

But then again, this might well make the whole problem worse for those that don't want so much hanging on the roll of the dice and prefer the assurance of the unmoving modifier.
Or then again again, maybe the proficiency die is way more popular than i'm led to believe, and this wouldn't be necessary anyways.

Anyways, need more sleep and less coffee.

[Mastikator]

It would make ability score modifiers stronger. A +1d4 is ~2.5, 1.5 more than a +1. A +1d12 is ~6.5, 1.5 more than +5

A +1 should be a d2, a +2 is 1d4, a +5 is d12

However what do you do if your ability score modifier is a +6 or +7 (belt of giant strength, maxed out barb, tomes)? I know nobody with d14s and d16s. There's no store in my city that sells those.

[Bane's Wolf]

Using a proficiency die actually sounds very interesting.
I remember reading about it in the DMG, and i've heard that they actually tried it as a core mechanic during the DnDNext playtests before 5th Ed was released.

I'd like to give it a shot, but i DM mostly for complete newbies. I've come to realize that what i consider simple mechanics can be overwhelming for players with no experience.
I doubt i'll be trying it on them soon.

I'll definitely see if my more experienced friends are up for giving it a shot next time we play. I'm not an expert at statistics, but i kinda feel like it would lead to the occasional EXTREME SUCCESS!, especially if you add bless and bardic inspirations to a roll :smallbiggrin:

[MoiMagnus]

Had a late night thought about this seemingly uncommon DMG optional rule. I'm guessing (stabbing in the dark really) that people don't like it because it increases random swing of a roll in a situation where the d20 already swings plenty enough already,

It doesn't, doesn't it?
d20+d4 is less swingy than d20+2.
Sure, the range is bigger, but the curves already reduces the variance.

The main reason why peoples don't like it is that it reduces "pre-computation" and increase "post-computation". And by pre-computation I mean every computation that you do once when level up and then never have to do it again, and by post-computation I mean any addition that you need to do after rolling dice. Currently, you can make the sum of proficiency + ability + equipments (like a sword +2 or whatever) and write the result on your character sheet, and then when you make an attack you can just roll a d20 and add this number. While if you have a proficiency die the computation when you roll dice is the sum of three numbers instead of a sum of two numbers.

Sure, it's not much, but when you do 3 attacks with advantage per round, it starts to be significant on mental load, so it needs to add something to the game to be worth it.

On the other hand, bless being a d4 is actually better for mental load. As since it is a circumstantial bonus, you can't pre-compute the +2 on your character sheet (unless you have a setup with a low-level follower of yours casting bless on the party, but if you're the kind to have followers, you're not the kind of player to complain about mental load), so having a d4 which materially represents the blessing can help you to not forget about it.

[Gryndle]

My objection to using "proficiency dice" and the dice like Kane0 suggests is not about the effect it has on math. The problem I have with it is that one of 5E's major selling points to my group is lack of complexity.

We played Alternity, which was an awesome skill driven d20 system. EXCEPT for the fact that it used "situation dice" to modify your d20 rolls, ranging from a single d4 to multiple d20's. Which completely butchered the flow of combat.

Kane0's suggestion brings back nightmares of Alternity's situation dice, adding complexity for complexity's sake alone.

In my opinion, IF 5E needs any complexity added at all, it definitely is not with the die mechanics.

[KorvinStarmast]

Had a late night thought about this seemingly uncommon DMG optional rule. We used it. In one game. You end up rolling two dice instead of 1, and they are each different dice. On something like Roll20 you can set up a macro or a script to make it easy, and the quick addition isn't that hard mentally.

I disagree that attribute bonus being rollable helps the game. It's one of those fiddly bits that doesn't need to be added.

TBH, making the ability score bonus variable is something you could try to do ... how much extra random do you want in your game?

Anyways, need more sleep and less coffee. That depends on the time of day. :smallsmile:

My objection to using "proficiency dice" and the dice like Kane0 suggests is not about the effect it has on math. The problem I have with it is that one of 5E's major selling points to my group is lack of complexity. Likewise. The attempt to apply the KISS principle was one of 5e's features that I liked a lot.

[strangebloke]

It doesn't, doesn't it?
d20+d4 is less swingy than d20+2.
Sure, the range is bigger, but the curves already reduces the variance.
Yeah, basically.

More dice lowers the amount of variance within the range of possible results, but something like 1d20+1d4+1d6 has a much wider range of possible outcomes. If you use 3d6 instead of a d20 for example you're using way more dice and its a smaller range, and the resulting variance is quite low. I've tried that before and imo it becomes problematic as high AC enemies basically become untouchable and low AC enemies may as well have no AC at all.

Booting up Anydice, we get

mean 1d20+5 - 15.50
dev 1d20+5 - 5.77

mean 3d6+5 - 15.50
dev 3d6+5 - 2.96

mean 1d20+1d4+1d6 - 16.50
dev 1d20+1d4+1d6 - 6.12

so overall yeah, unless you really care about that effective +1 or -1 caused by dice not quite matching up with +X mods they're basically the same. And if you really care about that slight change in value you could account for it. In this case use a d3 and d6 or a pair of d4s.

[JellyPooga]

What if all static modifiers to a roll were turned into additional dice? Not d20+proficiency dice+ability dice, etc. but more "calculate your total mod and turn it into dice". E.g. if you had a total modifier of +9 (let's say +3 proficiency, +4 ability score, +2 from a magic item), you might roll 1d20+1d12+1d6. Now in order to alleviate issues assossiated with the bell curve, how might it affect play to make that exclusive rather than additive roll? i.e. roll dice and pick the highest, rather than add them up. Further, what if we included the option to break down your static modifier into *your choice* of additional die? e.g. instead of the d20+d12+d6 from earlier, the player chose to roll 1d20+3d6?

Just some thoughts.

[Mastikator]

Yeah, basically.
I've tried that before and imo it becomes problematic as high AC enemies basically become untouchable and low AC enemies may as well have no AC at all.


Turn AC into dice as well. Leather gives 10+d2 + dex mod dice, studded is 10 + d4 + dex mod dice, scale is 10 + d8 + dex mod (max d4). Chainmail is 10 + d12, splint mail is 10 + 2d6, plate is 10 + d6+d8. Turn shield into a d4 ac bonus.
You roll your AC every time you take a hit.
Your DC for spells and abilities could use this too. DC is 8 + proficiency dice + spellcasting ability dice, they roll d20 + proficiency dice (if proficient) + ability score mod dice.

Pure chaos on the table. :smallbiggrin:

[Joe the Rat]

It would make ability score modifiers stronger. A +1d4 is ~2.5, 1.5 more than a +1. A +1d12 is ~6.5, 1.5 more than +5

A +1 should be a d2, a +2 is 1d4, a +5 is d12

However what do you do if your ability score modifier is a +6 or +7 (belt of giant strength, maxed out barb, tomes)? I know nobody with d14s and d16s. There's no store in my city that sells those.

Gamescience and Impact! Miniatures have sets with 14s and 16s (and 3s and 5s and 24s). Also try "DCC Dice" as a google search.


As an aside, Dungeon Crawl Classics (DCC) actually does this for their martial classes: Warrior and Dwarf. Instead of a fixed attack bonus, they get a deed die (starting at d3, and grows with level). In addition to bigger to-hit bonuses, rolling 3 or more allows a Mighty Deed of Arms to be added to the action - basically stunting or battle-mastering on any good hit. This could be something to leverage for proficiency dice - hitting a set value garners additional benefits, provided you hit your target. instead of a sliding DC for better results, hitting 4+ on proficiency gets you an extra bit of info, a faster completion, a better product... whatever makes sense for the skill in question. Fixing that to the proficiency die only then gives proficiency (focus or specialty in an area) a benefit over attribute (general ability in a broad domain) without restricting the option to roll.



...And now I want to rework the system as an unholy combination of 5e, Savage Worlds, and AEG's roll-and-keep.

[Segev]

If you go so far as to change static bonuses to bonus dice on damage, you either complicate the mental rule on what doubles on critical hits, or you increase critical hit damage by doubling dice on what used to be static values (which aren't doubled).

Using proficiency dice and the like has the advantage of requiring less up-front mental math in terms of, "wait, what is my bonus again?" and allows you to try to make the mental effort simpler by just picking up more dice, but in the end, you still sum all the numbers together, and you still run into having to check which dice you get, so I don't think it really simplifies anything.

The higher variance makes things less certain - even expertise becomes +2d4/2d6/2d8/etc. and thus could add as little as +2 to your roll - but also enables higher highs, so more people CAN make those static checks that go above 20.

Ultimately, I think bounded accuracy is better served in both directions by not having more dice added to it. Having those come from spells like guidance makes the spells feel more special and also keeps the added variance low (since guidance doesn't scale, and you don't get lots of spells like it stacking up).

[Snails]

Ultimately, I think bounded accuracy is better served in both directions by not having more dice added to it. Having those come from spells like guidance makes the spells feel more special and also keeps the added variance low (since guidance doesn't scale, and you don't get lots of spells like it stacking up).

I agree, in spades. There is a simple feel to the 5e game where you usually just add your prof and stat mods to the d20, and that alone. Those numbers are not changing very often. Nor are you likely to be swapping around magic swords or whatever. Then it is a d4 for Bless or Guidance. That is it.
Simple. Easy to remember.

Adding the complexity of more dice would probably be an enjoyable experience for me. I am that kind of guy. But I do recognize that about a third of the people I play with are not that kind of guy. This is a purely negative change to the game for them.

[CapnWildefyr]

Meh.
I agree with those who said 'too many dice.'
Also, the variability of added dice also permits higher rolls. There's always going to be someone who rolls lucky and, at first level, hits even non-proficient skill rolls with a 25-28. The averages work out, Avg d20+3 = 14.5, Avg d20 + d8 = 15..., but the range does not. Personally I prefer simpler here, with less range. Keeps higher level PCs a little more special. Wanna have a chance to pick that DC25 lock? Gain a few levels first.

[strangebloke]

Meh.
I agree with those who said 'too many dice.'
Also, the variability of added dice also permits higher rolls. There's always going to be someone who rolls lucky and, at first level, hits even non-proficient skill rolls with a 25-28. The averages work out, Avg d20+3 = 14.5, Avg d20 + d8 = 15..., but the range does not. Personally I prefer simpler here, with less range. Keeps higher level PCs a little more special. Wanna have a chance to pick that DC25 lock? Gain a few levels first.

I'm a huge fan of bounded accuracy in general, but I think that difficulty is a little too bounded relative to people's ability to boost their results. Especially for ability checks. Even a level 1 party can end up with something like advantage + guidance + bardic inspiration + expertise for a combined total of 13 with advantage.

And there's definitely a subset of player that will lobby hard for more chances to make rolls if they think there's a chance to get lucky and do the 'impossible' thing. It's more a social problem than a mechanical one but this would make it worse.

Agreed about it being too much work. Rolling twice as many dice, doing twice as much small addition... not great for combat speed. I'd do this for a one shot though.

[Segev]

I'm a huge fan of bounded accuracy in general, but I think that difficulty is a little too bounded relative to people's ability to boost their results. Especially for ability checks. Even a level 1 party can end up with something like advantage + guidance + bardic inspiration + expertise for a combined total of 13 with advantage.

I don't actually see this as a defect. Rolls should always be consequential enough that, if you're rolling, and you fail, the failure has impact. So you shouldn't be fishing with lots of rolls for the high-end result, because it shouldn't be worth it. The other side of the coin is that I have no problem with a level 1 PC making a DC 30 if he's got all that stuff stacked up and rolls high on the d20. Good for him! It's "nigh-impossible," sure, but look at all the heroic effort that went into achieving it.

[Kane0]

Well that seems a fair concensus, although I admit 'dice everything!' sounds fun for a tier 1 oneshot.

[strangebloke]

I don't actually see this as a defect. Rolls should always be consequential enough that, if you're rolling, and you fail, the failure has impact. So you shouldn't be fishing with lots of rolls for the high-end result, because it shouldn't be worth it. The other side of the coin is that I have no problem with a level 1 PC making a DC 30 if he's got all that stuff stacked up and rolls high on the d20. Good for him! It's "nigh-impossible," sure, but look at all the heroic effort that went into achieving it.

Like I said its mostly a social issue. Lots of skill checks have to be adlibbed by the DM and players are prone to putting pressure on the DM to allow skill checks that shouldn't be allowed / that the DM doesn't have thought out well. The perception of something being possible is in some cases more disruptive than the actuality of being possible. Sure, if you're a good DM you can deal with it, come up with consequences etc. but the game really doesn't do a good job explaining what a DC 30 check should even allow.

Well that seems a fair concensus, although I admit 'dice everything!' sounds fun for a tier 1 oneshot.
Every rolls for race, class, traits, bonds, ideals, and stats in order. Proficiency bonus is replaced by a d4 or d6. Ability Scores are simulated via dice as well.

Let the chaos reign.

[Segev]

but the game really doesn't do a good job explaining what a DC 30 check should even allow.

I concur that doing this better would be desirable.

[animorte]

Going to toss my thoughts in here. I've mentioned somewhere recently that I don't care much for growing dice. 1d4 eventually up to 1d12 (or 20) still gives me the option to land on a 1. If I'm hoping to be proficient, I don't much like the idea of randomly being pathetic. If the base is a 1d20 + whatever else you're adding to it, that's already a possible crap roll.

I much prefer stacking dice. Start out with a 1d4 for proficiency then bump that to a 2d4, then 3d4, 4d4, etc. This way my minimum grows as well, whatever the actual base dice may be.

Of course there are still wild magic tables and several other things that you are just rolling for different options and it's not strictly a power/skill number.

[MoiMagnus]

Going to toss my thoughts in here. I've mentioned somewhere recently that I don't care much for growing dice. 1d4 eventually up to 1d12 (or 20) still gives me the option to land on a 1. If I'm hoping to be proficient, I don't much like the idea of randomly being pathetic. If the base is a 1d20 + whatever else you're adding to it, that's already a possible crap roll.

I much prefer stacking dice. Start out with a 1d4 for proficiency then bump that to a 2d4, then 3d4, 4d4, etc. This way my minimum grows as well, whatever the actual base dice may be.

Thinking about your comment, I've come up with a weird table to handle proficiency with more dice while maintaining the maths:

[Untrained] d20 => d20
[Lv 01-04] d20+2 => 2d12
[Lv 05-08] d20+3 => 3d8
[Lv 09-12] d20+4 => 4d6
[Lv 13-16] d20+5 => 6d4
[Lv 17-20] d20+7 => 7d4

The expected value is somewhat preserved: 10.5/13/13.5/14/15/17.5 instead of 10.5/12.5/13.5/14.5/15.5/16.5
The minimum is always increasing, at the cost of the maximum remaining mostly constant (so in some way, it's even more "bounded accuracy" than official 5e).
Obviously, that's a lot of dice at high level, and definitely too many d4s to match my tastes.

[Saelethil]

I’ve considered using it but I think that as written it would result in slower gameplay for minimal improvement. I do think that the scaling would be good for any feature with scaling dice (monk damage, superiority dice, bardic inspiration, Rage Damage) and I would be open to Expertise adding a PD instead of doubling your PB.

[Yakk]

It doesn't, doesn't it?
d20+d4 is less swingy than d20+2.
Sure, the range is bigger, but the curves already reduces the variance.
No, that is wrong.

1d20+1d4 has higher variance than 1d20+2 -- it is swingier.

More dice = less swing rule applies *when the total and the average is similar*.

Here, we are adding a "d always 2" vs a "1d4", and adding swingy makes things more swingy. Dividing a big die into smaller dice makes it *less* swingy.

Var(d20) is 399/12, var(d4) is 15/12. Var(d20+d4) is 414/12 (it adds), and Var(+2) is 0.

Now, the difference here is small, because almost all of the variance comes from the d20.

But Var(d12) is 143/12 -- here, Var(d20+d12) = 542/12, which is noticeably larger than Var(d20+6) = Var(d20) = 399/12.

---

I don't particularly like the "replace attributes with dice" for a number of reasons mentioned above.

But, as an aside, one thing I've played with is replacing the "your strength is 19" type items with ones that basically give you such a die.

So you'd get a 1d4 "might" die with belt of hill giant strength that adds to strength damage rolls and strength attribute checks... but not strength accuracy checks!

[Hytheter]

The notion that replacing a static number with a dice makes the results less random is laughable. Stop obsessing with bell curves and apply some common sense, people.

[Yakk]

The notion that replacing a static number with a dice makes the results less random is laughable. Stop obsessing with bell curves and apply some common sense, people.
Sure, but which is more random, 2d10+2 or 1d20+1d4?

At some point you might as well learn basic stats if you actually care. And Var(1dX) = (X^2-1)/12 isn't hard to remember.

You can ignore the -1 and /12 part if you really want, and know that randomness roughly the sum of the squares of the die sizes. In fact, for SD, you take the square root.

This ends up looking like distance. Which is longer, a 20 unit long line, or one that goes 10 in one direction, 10 in another, and 4 in the 3rd? The square root of the sum of the squares is very close to proportional to the standard deviation of a collection of dice.

[stoutstien]

The notion that replacing a static number with a dice makes the results less random is laughable. Stop obsessing with bell curves and apply some common sense, people.

Because numeracy tends to act contraire of "common sense" and has its own language to contend with.

[strangebloke]

The notion that replacing a static number with a dice makes the results less random is laughable. Stop obsessing with bell curves and apply some common sense, people.

I mean its a broader range of possibilities but a tighter variance. Sure, sometimes you'll get someone getting a crazy 35 on an attack roll, but most of the time you'll have results very similar to what you already have.

as an example, which is more random: A greatsword or a greataxe? The greatsword has twice the dice, but the results are way less random.

[Hytheter]

I mean its a broader range of possibilities but a tighter variance.

No it's not. Yakk already demonstrated that... And so did you!


mean 1d20+5 - 15.50
dev 1d20+5 - 5.77

mean 3d6+5 - 15.50
dev 3d6+5 - 2.96

mean 1d20+1d4+1d6 - 16.50
dev 1d20+1d4+1d6 - 6.12

Notice how adding dice to the d20 instead of modifiers increases the deviation (which is just the square root of variance)? The probability distribution isn't linear anymore but it's not enough to compensate for the expanded range.


as an example, which is more random: A greatsword or a greataxe? The greatsword has twice the dice, but the results are way less random.

That's entirely irrelevant to my point, since it is not 'replacing a static number with a dice'. Yes, it is well established that splitting up one dice into a number of smaller dice reduces the variance, hence 2d6 is less random than a d12. The problem is people are taking that notion too far and just assuming "more dice = less swingy" as a universal principle, leading to claims like "d20+d4 has less variance than d20+2" which is just not true. (It's not even close to a bell curve, either, for the record.) All else being equal, adding more randomisers to a function will always make things more random, not less. Go figure.


d6 1.71
2d6 2.42
3d6 2.96
d20 5.77
d20+d4 5.87
d20+d6 6.01
d20+d8 6.20
d20+d10 6.44
d20+d12 6.72
d20+d20 8.15
All Dice 7.93

edit:


Sure, sometimes you'll get someone getting a crazy 35 on an attack roll, but most of the time you'll have results very similar to what you already have.

I was meant to address this too.
On a d20+d4, you have a 6.25%* chance of rolling a result that should otherwise be impossible (vs d20+2), which is not negligible. By the time you get up to d20+d12, that increases to 15% (vs d20+6). Oh, but we need to add some ability dice, too? d20+d4+d6 gives 9.16% chance of an impossiblility (vs d20+5) and d20+2d12 gives a whopping 22.57% chance of impossibility (vs d20+12) - that's almost one in four rolls being outside the normal realm of possibility!

edit2: *5%. I had it right, then I double-checked before posting and got it wrong. Oops...

[Yakk]

Now, your standard deviation *as a fraction of your range* goes down with more dice. The range of 1d4+1d20 is (20+4)-(1+1) = 22, while the range of 1d20+2 is 19.

SD/Range of 1d20+1d4 is sqrt((20^2-1 + 4^2 - 1)/((20-1)+(4-1))^2/12) = 0.267
SD/Range of 1d20+2 is sqrt((20^2-1)/(20-1)^2/12) = 0.303

Intuitively, this means that the results of d20+2 are more spread out over its range than the results of 1d20+1d4. But the absolute spread is higher for 1d20+1d4.

[verbatim]

I've thought about trying this on roll20 before where all rolling can be automated away, but the extra dice rolling feels like an uneccesary pain irl.

[Pex]

Way back when during the 2E era I learned of the idea of changing +# weapons to dice. A +1 weapon is a +1d4 to hit and damage, +2 is 1d6, etc. It was so long ago I do forget the details. The extra die may have only been for damage keeping the normal +# to hit. Point is as proposed you roll dice instead of using static numbers. I tried it as an experiment when running a one-shot at a convention. I found it added to the length of combat. You needed to look at separate dice, add separate dice, it adds up over time. It didn't run smoothly, and the combats were getting boring. I just ended a combat early, the PCs won, and we moved on without using it anymore. Players do like rolling dice, but too many too often boggles the game. Even in games now I constantly see players forget about the +1d4 for Bless, DMs forgetting the -1d4 for Bane (not on purpose). Players like rolling 8d6 fireball damage, but DMs will resort o average damage from monsters even on normal attacks because they don't to keep rolling so many dice for so many monsters. Not all DMs of course, but it happens and I'm not disparaging the practice just saying it happens.

Long story short (too late) this is an idea to me that looks good on paper but fails in practice.

[strangebloke]

No it's not. Yakk already demonstrated that... And so did you!

Notice how adding dice to the d20 instead of modifiers increases the deviation (which is just the square root of variance)? The probability distribution isn't linear anymore but it's not enough to compensate for the expanded range.

I mean if you look at my original post, my statement was: "More dice lowers the amount of variance within the range of possible results, but something like 1d20+1d4+1d6 has a much wider range of possible outcomes."

Which maybe isn't the right way to put it, but yeah.

Completely fair to point out that it ends up being wildly chaotic at high levels.

[Mastikator]

I ran a simulation and collected some data.

I rolled a million times.

Scenario 1) 1d20 + 4 (proficiency bonus) + 5 (ability score bonus)

Scenario 2) 1d20 + 1d8 (proficiency dice) + 1d10 (ability score dice)

I then counted the results for each scenario and placed it on two graphs. I then screenshotted the graphs and placed them on top of each other.

https://i.imgur.com/jIW4v2n.png

Results: The proficiency dice added on average +1 on the result.
Rolling normally is 5% on each possible value (as expected). However rolling with proficiency dice and ability score dice creates a big probability space of getting significantly lower, 3 is always the lowest possible value. And while you can get a higher value it becomes increasingly unlikely.
Rolling a total value of 29 is 5% with a normal roll but much lower with d20 + d8 + d10 because ALL the dice must roll high.

Edit-

I switched to a bar graph and made it bigger. Also increased to 2 million dice rolls to decrease noise.

https://i.imgur.com/sid0ZmZ.png

[Hytheter]

I ran a simulation and collected some data.

I rolled a million times.

Scenario 1) 1d20 + 4 (proficiency bonus) + 5 (ability score bonus)

Scenario 2) 1d20 + 1d8 (proficiency dice) + 1d10 (ability score dice)

I then counted the results for each scenario and placed it on two graphs. I then screenshotted the graphs and placed them on top of each other.

https://i.imgur.com/jIW4v2n.png

Results: The proficiency dice added on average +1 on the result.
Rolling normally is 5% on each possible value (as expected). However rolling with proficiency dice and ability score dice creates a big probability space of getting significantly lower, 3 is always the lowest possible value. And while you can get a higher value it becomes increasingly unlikely.
Rolling a total value of 29 is 5% with a normal roll but much lower with d20 + d8 + d10 because ALL the dice must roll high.

You either did something wrong in your process or you got a real statistical fluke*, because that distribution is rather abnormal with a strange drop on the higher end where it should be symmetrical. The actual chance of rolling a 29 on d20+d10+d8 is 3.25%; somewhat lower than 5% but much higher than your graph suggests.

*Which is entirely possible. The 'roll a bunch of dice' method of determining probability is unreliable for this reason. It's a lot easier and more accurate to just use AnyDice.

[Mastikator]

Yes I did. I rolled the rolled results of the proficiency and ability bonus LMAO

https://i.imgur.com/6B9FSLq.png

Here are the corrected results of 2 million dice rolls.

[Hytheter]

Yeah, I was gonna say after seeing the edit, if that's 2million rolls there's definitely some kinda mistake! Graph looks a lot more normal (pun not intended) now.

I'd still recommend using AnyDice for stuff like this, though.

[Snails]

Scenario 2) 1d20 + 1d8 (proficiency dice) + 1d10 (ability score dice)

I may not be as handy at making beautiful graphs as you are, but I do have a little program that makes precise calculations for these things...

Distribution of values for : 1d20+1d10+1d8
Distribution:
3 --> 0.06% (1 of 1600)
4 --> 0.19% (3 of 1600)
5 --> 0.38% (6 of 1600)
6 --> 0.63% (10 of 1600)
7 --> 0.94% (15 of 1600)
8 --> 1.31% (21 of 1600)
9 --> 1.75% (28 of 1600)
10 --> 2.25% (36 of 1600)
11 --> 2.75% (44 of 1600)
12 --> 3.25% (52 of 1600)
13 --> 3.69% (59 of 1600)
14 --> 4.06% (65 of 1600)
15 --> 4.38% (70 of 1600)
16 --> 4.63% (74 of 1600)
17 --> 4.81% (77 of 1600)
18 --> 4.94% (79 of 1600)
19 --> 5.00% (80 of 1600)
20 --> 5.00% (80 of 1600)
21 --> 5.00% (80 of 1600)
22 --> 5.00% (80 of 1600)
23 --> 4.94% (79 of 1600)
24 --> 4.81% (77 of 1600)
25 --> 4.63% (74 of 1600)
26 --> 4.38% (70 of 1600)
27 --> 4.06% (65 of 1600)
28 --> 3.69% (59 of 1600)
29 --> 3.25% (52 of 1600)
30 --> 2.75% (44 of 1600)
31 --> 2.25% (36 of 1600)
32 --> 1.75% (28 of 1600)
33 --> 1.31% (21 of 1600)
34 --> 0.94% (15 of 1600)
35 --> 0.63% (10 of 1600)
36 --> 0.38% (6 of 1600)
37 --> 0.19% (3 of 1600)
38 --> 0.06% (1 of 1600)

BTW: I can also calculate opposed checks, i.e. "what are the odds of 1d20+5 equalling or exceeding 1d20+3+1d4". Or weirder things "3d10 pick highest 2".

[Yakk]

Mastikator, graph the cumulative distribution function.

In D&D, "rolling exactly X" isn't useful information, which you graphed. You care about "rolled X or better".

You are, in effect, graphing the derivative of the graph we really care about. And when you do that you end up having a bunch of noise we shouldn't care about showing up as super-important.

Also, note that there are (20)*(12)*(10) = 2400 total equally likely rolls. Doing a 2 million die simulation is both inaccurate and a lot more work than calculating the exact probabilities!

https://anydice.com/program/2db50 with "at least" compares the two side by side.

You can see the +1 average difference from the gap between the two.