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

[Ovinomancer]

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]

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]

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.

Spoiler

   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).

Spoiler

   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

 

[Esker]

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]

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.

Spoiler

   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).

Spoiler

   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]

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.

 

[Ovinomancer]

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.

:headdesk:

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]

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]

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.

 

[Esker]

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.