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

[Ovinomancer]

@Ovinomancer, here's an analogy for you. I'm curious what your intuitions are.

Suppose all this time 5e had used a percentile system to resolve checks, instead of the d20, and you had to roll at or under the DC to succeed. A medium difficulty task was DC 60, a really hard task was DC 10, etc. The proficiency bonus started at +20 and went up in increments of 10, and ability scores went from 0 to 100, with modifiers set to 0 at a 50 and going up by 10 whenever the ten's digit goes up, so, when you hit 60 you're at +10, 70 is a +20, etc., all the way up to +50 at 100. Also, instead of adding your bonus to the roll, you added it to the DC (thus making it easier to succeed).

Now someone comes along and says, "It's a pain to have to roll two dice for every check, and also wouldn't it be nice if the DM could keep some DCs secret without having to know everybody's bonuses? What if we scaled ability scores and bonuses down by a factor of 10, rolled a d10 instead of a d100 to resolve outcomes, added bonuses to the roll instead of the DC, and said that a success was rolling at or above the target instead of at or below it? To keep things comparable, we'll modify all the DCs to be DC' =1 + (100 - DC)/10, so 60 becomes 5, 50 becomes 6, 40 becomes 7, and so on."

First question for you: if nobody ever used DCs that weren't multiples of 10, would this change have any effect on the outcomes in the game? (I'm not asking whether it would have an effect on how much work it is, just whether it would affect outcomes)

Second question: Suppose somebody objected to this change, saying: "You can't say that this won't have an impact! We used to have 100 increments, and now we only have 10!"

The designer shows the objector a line graph, with two different sets of labels on the x-axis: The first set of labels go from 0 to 100, representing DCs in the old (percentile) system. The second shows the corresponding DC in the new system: 0 is aligned with 11, 5 is aligned with 10.5, 10 is aligned with 10, 20 with 9, 30 with 8, etc. Then there are two lines. The one for the old system shows that a DC 50 check has a 50% success rate, a DC 55 has a 55% success rate, a DC 60 check has a 60% success rate, etc. The second only has points at whole numbers, but at those spots, lines up with the first one.

"Nobody uses DCs that aren't a multiple of 10," they say. "The graph lines up where it matters."

Supposing it's true that DCs are always multiples of 10, who is right?

Third: Suppose the objector were a DM who actually liked to use DCs in multiples of 5. They approach the redesigner, red-faced, saying: I have a DC 65 check, which worked perfectly well before, but now you're telling me the DC is 4.5! You can't roll a 4.5!"

"You're right," says the designer. "How about this: round your DCs down to the nearest whole number, but keep track of whether it was a half originally. So your 4.5 becomes a 4. But if the player rolls exactly 4 on their d10 (after modifiers), have them then roll a d6. If they get 4 or more, they succeed, otherwise they fail."

They then go to their graph, and fill in points on the second line at 10.5, 9.5, etc., which sit at 5% success, 15% success, etc., explaining, "Your DC 65 check becomes almost like a DC 4 check, except it's a little more difficult because there's an extra step involved to succeed. A DC 4 check has a 70% chance of success, since you can roll anything but a 1, 2 or 3. In your case, they have one extra way to fail: by rolling a 4 and then rolling a 1, 2 or 3 on the d6. That happens (1/10) * (3/6) of the time, or 5%. So there's now a 35% chance of failing, and a 65% chance of succeeding, just like there would have been before.

The objector thinks for a minute and says, "That's a B.S. kludge. Those points you're drawing don't exist! You can't just say that you can have a DC 4.5 check, if you can't roll 4.5! What kind of statistics mumbo jumbo is this?"

Is the designer pulling a fast one? Does their suggested fix allow for 55 or 65 DCs, etc. to work as intended? Or is something wrong?

This hypo just shows you haven't grasped my argument at all. If you scale both the resolution method and the target numbers, fine, you've done the same things to both sides. The problem would occur if you change the scale on only one side and then compare to a completely different resolution at the original scale to claim similarity, you've done something wrong.

This is the problem, as I explain it again.

 

[Esker]

This hypo just shows you haven't grasped my argument at all. If you scale both the resolution method and the target numbers, fine, you've done the same things to both sides. The problem would occur if you change the scale on only one side and then compare to a completely different resolution at the original scale to claim similarity, you've done something wrong.

This is the problem, as I explain it again.

I posted the hypothetical for the purposes of removing complexity and isolating one aspect of the comparison at a time. Discovering that you are fine with that hypothetical, where we've rescaled DCs and rolls, helps us narrow down the source of your discomfort, by ruling out the unequal spacing as a cause (since a d10 and a d100 have unequal spacing). It also confirms that you're comfortable using a confirmation die or similar mechanism to fill in the loss in distinctions between consecutive DCs created by unequal spacing (as I do when comparing ordinary 1d20 to 2*3d6-10).

The only differences between that hypothetical and the original scenario (well, there are two original scenarios, differing by how much of a role you want luck to play, but let's focus on the 1d20 vs 2*3d6-10 one for now, since one of them is vanilla and thus hopefully has good intuition behind it, which should be easier than comparing two unfamiliar schemes to each other).

In the 1d20 vs 2*3d6-10 case, we're rolling against the same DCs, so that removes yet one more point of complexity that we have to deal with in the 3d6 vs rescaled 1d20 scenario.

Ok. So, with a regular 1d20, the distinct (adjusted) DC ranges are: [-infinity to 1], 2, 3, ..., 20, and [21 to infinity]. There are 21 functionally distinct DCs here. Anything with an adjusted DC less than 1 is functionally equivalent to an adjusted DC of 1, since there are no additional ways to succeed on, say, an adjusted DC of 0 that don't also succeed on an adjusted DC of 1.

With 2*3d6-10, on its own, we have the following sets of DCs that can be distinguished: [-infinity to -4], {-3,-2}, {-1,0}, {1,2}, ..., {21,22}, {23,24}, {25,26}, [27 to infinity]. There are only 17 of these, and they don't line up with the ones in the 1d20 system. Because we can't roll odd numbers, -2 is no harder than -3, 0 is no harder than -1, etc. The confirmation mechanic breaks this up further (just as the confirmation mechanic in the d10 system in the hypothetical): since the confirmation mechanic only affects even DCs, we can now distinguish the following sets: [-infinity to -5], -4, -3, -2, -1, 0, 1, ..., 25, 26, [27 to infinity]. We've actually created more distinctions than we had with 1d20 -- a total of 33 -- because [-infinity, 1] is subdivided into seven different sets, as is [21 to infinity].

Is this a problem? Well, it depends on your tolerance for approximation. In the 1d20 system, every DC from -infinity to 1 has the same difficulty, whereas in the 2*3d6-10 system, we get slight increases in difficulty when we go from -5 to -4, -4 to -3, from -3 to -2, etc., because we are losing ways to succeed: rolling a 3 on the 3d6 definitely succeeds if the DC is -5 or less, but only a 50% chance if the DC is -4, and 0% if it is -3. Rolling a 4 definitely succeeds if the DC is -3, but it only has a 50% chance of succeeding if it is -2, and a 0% chance if it is -1. And so on.

There are two ways to look at the effects of this discrepancy. First, we can ask what happens if you have two characters facing a DC of 1, but one is rolling d20 and the other is rolling 2*3d6-10-(d2-1). The first character is guaranteed to succeed; they don't even need to roll. The second might fail, because they might roll a 5 or below on the 3d6 (corresponding to 0 or below after the transformation). This has a 4.6% chance of occurring. I'm not pretending this is nothing --- it's not, clearly, as it's only slightly less likely than rolling a 1 on the d20. But this is actually the worst the comparison ever gets. At DC 0 they still might fail (whereas the 1d20 character can't, obviously), by rolling a 3 or 4, or rolling a 5 and failing their confirmation roll. But this is less likely. And so on. The same thing happens on the other end of the spectrum, at those very high adjusted DCs.

The other way you can look at the discrepancy is in terms of the value of a +1. If you have a character whose bonus puts them at an adjusted DC of 1 on a particular check, we can ask, what is the impact if that character gets an additional +1? Well, regardless of the roll mechanic, that +1 reduces their adjusted DC to 0. Now, if they're using a d20 that does nothing; they were already at 100% success. If using 2*3d6-10 though, it gives them a little bit of a boost: about a 1.4 percentage point increase in their success chance (i.e., the probability of both rolling a 5 and succeeding in confirming).

Here, the graphs don't look nearly so similar, but the actual magnitudes of the discrepancies are still pretty small. The worst discrepancy in the value of a +1 is about 2.5 percentage points, which happens if you're currently sitting at a DC of 2: with 1d20 a +1 is always worth 5% within the range of 2 to 21 (because at each of these we add a new way to succeed), but at DC 2, a +1 is only worth about half that. The same at DC 21. Here are the graphs (since you didn't like the fact that I was interpolating between points before, I'm just plotting the points this time):

 

[Ovinomancer]

I posted the hypothetical for the purposes of removing complexity and isolating one aspect of the comparison at a time. Discovering that you are fine with that hypothetical, where we've rescaled DCs and rolls, helps us narrow down the source of your discomfort, by ruling out the unequal spacing as a cause (since a d10 and a d100 have unequal spacing). It also confirms that you're comfortable using a confirmation die or similar mechanism to fill in the loss in distinctions between consecutive DCs created by unequal spacing (as I do when comparing ordinary 1d20 to 2*3d6-10).

You assume too much. I'm fine with a rescaled system of resolution and targets within that system ONLY. Once you begin comparisons, the gaps become important.

And, the kludge is still a kludge -- it's an grafted on mechanic to correct an failure in the original system. It's not clever, and adding a kludge is admitting the original system failed so you need another system on top of it to try to correct your failure. The problem with your kludge is that you're using it to address target numbers that don't exist in the scaled schema, only when you try to compare to a different schema.

The only differences between that hypothetical and the original scenario (well, there are two original scenarios, differing by how much of a role you want luck to play, but let's focus on the 1d20 vs 2*3d6-10 one for now, since one of them is vanilla and thus hopefully has good intuition behind it, which should be easier than comparing two unfamiliar schemes to each other).

In the 1d20 vs 2*3d6-10 case, we're rolling against the same DCs, so that removes yet one more point of complexity that we have to deal with in the 3d6 vs rescaled 1d20 scenario.

This, right here, is the error. You are NOT using the same DCs in each system. The scaled system uses DCs stepped by 2, because it's scaled. The d20 isn't. You cannot compare these thing without making an error, because possibilities exist for one that do not for the other.

Ok. So, with a regular 1d20, the distinct (adjusted) DC ranges are: [-infinity to 1], 2, 3, ..., 20, and [21 to infinity]. There are 21 functionally distinct DCs here. Anything with an adjusted DC less than 1 is functionally equivalent to an adjusted DC of 1, since there are no additional ways to succeed on, say, an adjusted DC of 0 that don't also succeed on an adjusted DC of 1.

With 2*3d6-10, on its own, we have the following sets of DCs that can be distinguished: [-infinity to -4], {-3,-2}, {-1,0}, {1,2}, ..., {21,22}, {23,24}, {25,26}, [27 to infinity]. There are only 17 of these, and they don't line up with the ones in the 1d20 system. Because we can't roll odd numbers, -2 is no harder than -3, 0 is no harder than -1, etc. The confirmation mechanic breaks this up further (just as the confirmation mechanic in the d10 system in the hypothetical): since the confirmation mechanic only affects even DCs, we can now distinguish the following sets: [-infinity to -5], -4, -3, -2, -1, 0, 1, ..., 25, 26, [27 to infinity]. We've actually created more distinctions than we had with 1d20 -- a total of 33 -- because [-infinity, 1] is subdivided into seven different sets, as is [21 to infinity].

Yes, you can determine if you roll greater than a 2 in the 2*3d6 system and it's mathematically the same as the probability you roll a 3 or greater. However, 2 as a target number DOES NOT EXIST in the 2*3d6-10 system. This is the reification sin -- you confuse being able to create a probability for an event that does not exist in the system. Here, a target number of 2. You confuse that comparing a probability of greater than a number is not the same analysis as greater than or equal to, but you mix an match these to fool yourself into thinking 2's actually exist in the 2*3d6 system.

2 exists for d20, though, which is why you can't compare these systems. One has even DCs, the other doesn't (except below 0, which is an artifact of the recentering).

Is this a problem? Well, it depends on your tolerance for approximation. In the 1d20 system, every DC from -infinity to 1 has the same difficulty, whereas in the 2*3d6-10 system, we get slight increases in difficulty when we go from -5 to -4, -4 to -3, from -3 to -2, etc., because we are losing ways to succeed: rolling a 3 on the 3d6 definitely succeeds if the DC is -5 or less, but only a 50% chance if the DC is -4, and 0% if it is -3. Rolling a 4 definitely succeeds if the DC is -3, but it only has a 50% chance of succeeding if it is -2, and a 0% chance if it is -1. And so on.

There are two ways to look at the effects of this discrepancy. First, we can ask what happens if you have two characters facing a DC of 1, but one is rolling d20 and the other is rolling 2*3d6-10-(d2-1). The first character is guaranteed to succeed; they don't even need to roll. The second might fail, because they might roll a 5 or below on the 3d6 (corresponding to 0 or below after the transformation). This has a 4.6% chance of occurring. I'm not pretending this is nothing --- it's not, clearly, as it's only slightly less likely than rolling a 1 on the d20. But this is actually the worst the comparison ever gets. At DC 0 they still might fail (whereas the 1d20 character can't, obviously), by rolling a 3 or 4, or rolling a 5 and failing their confirmation roll. But this is less likely. And so on. The same thing happens on the other end of the spectrum, at those very high adjusted DCs.

The other way you can look at the discrepancy is in terms of the value of a +1. If you have a character whose bonus puts them at an adjusted DC of 1 on a particular check, we can ask, what is the impact if that character gets an additional +1? Well, regardless of the roll mechanic, that +1 reduces their adjusted DC to 0. Now, if they're using a d20 that does nothing; they were already at 100% success. If using 2*3d6-10 though, it gives them a little bit of a boost: about a 1.4 percentage point increase in their success chance (i.e., the probability of both rolling a 5 and succeeding in confirming).

When you recenter the mean of the method, you must recenter the mean of the DCs, or your system is very, very much not the same as what you started with. This is like saying that needing a 3 on 3d6 is the same as needing a 3 on 2*3d6-10. It's not. The same value on 2*3d6-10 as a 3 on 3d6 is -4. This is the other half of the fundamental reason you can't compare the systems as you're doing -- you're comparing values of DC that do not align but, because it graphs, you've fooled yourself into thinking it does.

Here, the graphs don't look nearly so similar, but the actual magnitudes of the discrepancies are still pretty small. The worst discrepancy in the value of a +1 is about 2.5 percentage points, which happens if you're currently sitting at a DC of 2: with 1d20 a +1 is always worth 5% within the range of 2 to 21 (because at each of these we add a new way to succeed), but at DC 2, a +1 is only worth about half that. The same at DC 21. Here are the graphs (since you didn't like the fact that I was interpolating between points before, I'm just plotting the points this time):

Dear god, but you've graphed two different PDFs on top of each other as if they're the same thing. You've graphed the PDF for greater than x on the half steps, and greater than or equal to on the whole steps. For someone that lectured on the basics of probability and made semantic arguments because I haven't gone jargon but tried to keep this jargon free, this must be an embarrassing error -- graphing two different probability questions on the same graph and pretending they're the same thing. And that doesn't even get to the other system you're graphing and the issues I've outlined above.

And, the worst discrepancy is still where I can roll 2*3d6-10 and can't roll a d20. Surely, this must sink in sometime? I'm losing hope. I'm sure the response will continue to not get the problem I've been rephrasing for many, many posts now -- you have different scales of both die outcomes AND DCs, but you're treating the DC scale as if it's the same. It is not.

 

[Esker]

This, right here, is the error. You are NOT using the same DCs in each system. The scaled system uses DCs stepped by 2, because it's scaled. The d20 isn't. You cannot compare these thing without making an error, because possibilities exist for one that do not for the other.

No. The set of DCs is the same. I am looking at trying to meet DCs in the range -5 to 27, and comparing apples to apples.

Yes, you can determine if you roll greater than a 2 in the 2*3d6 system and it's mathematically the same as the probability you roll a 3 or greater. However, 2 as a target number DOES NOT EXIST in the 2*3d6-10 system.

If I'm a rogue with a +12 to stealth and I need to beat a passive perception of 11, then my target on a d20 is 0. That's the minimum value I could roll and succeed. I can't actually roll 0, but that's still my target number. And I can still find the probability of getting 0 or better. It just happens to be the same as the probability of getting 1 or better.

You confuse that comparing a probability of greater than a number is not the same analysis as greater than or equal to

It's the same exactly when the probability of "equal to" is zero. I'm starting to think that you don't think it's valid to talk about events with probability zero; that the event doesn't exist or something? Is that what's happening?

When you recenter the mean of the method, you must recenter the mean of the DCs, or your system is very, very much not the same as what you started with. This is like saying that needing a 3 on 3d6 is the same as needing a 3 on 2*3d6-10. It's not. The same value on 2*3d6-10 as a 3 on 3d6 is -4.

No, I've subtracted 10 from the rolls so that I can compare same DC to same DC. You roll 3d6, multiply it by 2, and subtract 10. If the result after all of that is above your DC, you succeed. So when I consider a DC of -4, that's a DC of -4 for either method: a rogue with a +15 in thieves tools trying to pick a DC 10 lock. I need to roll a -4 or better to succeed. If I'm rolling 1d20 this is a guarantee, and it's the same difficulty as a target of 1. If I'm rolling 2*3d6-10 and have to confirm on ties, it's almost a guarantee, but not quite, since I could roll -4 exactly and then fail to confirm. But this is extremely unlikely.

Dear god, but you've graphed two different PDFs on top of each other as if they're the same thing. You've graphed the PDF for greater than x on the half steps, and greater than or equal to on the whole steps.

Nope, not what this shows. Actually none of the points are either of those things. If you read the paragraph above the plot, or even looked at the Y axis label, you'd see that all the points are the change in the probability of success if I'm trying to hit a DC X and I get an extra +1 to my roll (effectively making the DC X-1). This works out to be the probability of hitting X-1 exactly. You can relabel the Y axis P(roll = X-1). (That's why the d20 probabilities are 0.05 from 2 to 21 and not 1 to 20, since if I start at DC 2, I gain 5% success if I get +1, but if I start at DC 1, I gain nothing if I get an extra +1)

I'm graphing the same thing for both evens and odds. The reason there are points at odds on the 2*3d6-10 curve is that I'm really rolling 2*3d6-(d2-1), and so I really can hit both odd and even numbers.

And, the worst discrepancy is still where I can roll 2*3d6-10 and can't roll a d20.

I mean, it depends on how your measure the discrepancy. I'm actually measuring the discrepancy at those values (as simple difference in probability) and including it in the plot. Not sure why you think I haven't taken it into account. If you want to measure discrepancy as a ratio, well then yeah, it's big, since one of the terms is zero. But you've never in this whole massive thread suggested that your problem was in the way I was comparing probabilities.

 

[Ovinomancer]

No. The set of DCs is the same. I am looking at trying to meet DCs in the range -5 to 27, and comparing apples to apples.



If I'm a rogue with a +12 to stealth and I need to beat a passive perception of 11, then my target on a d20 is 0. That's the minimum value I could roll and succeed. I can't actually roll 0, but that's still my target number. And I can still find the probability of getting 0 or better. It just happens to be the same as the probability of getting 1 or better.



It's the same exactly when the probability of "equal to" is zero. I'm starting to think that you don't think it's valid to talk about events with probability zero; that the event doesn't exist or something? Is that what's happening?



No, I've subtracted 10 from the rolls so that I can compare same DC to same DC. You roll 3d6, multiply it by 2, and subtract 10. If the result after all of that is above your DC, you succeed. So when I consider a DC of -4, that's a DC of -4 for either method: a rogue with a +15 in thieves tools trying to pick a DC 10 lock. I need to roll a -4 or better to succeed. If I'm rolling 1d20 this is a guarantee, and it's the same difficulty as a target of 1. If I'm rolling 2*3d6-10 and have to confirm on ties, it's almost a guarantee, but not quite, since I could roll -4 exactly and then fail to confirm. But this is extremely unlikely.



Nope, not what this shows. Actually none of the points are either of those things. If you read the paragraph above the plot, or even looked at the Y axis label, you'd see that all the points are the change in the probability of success if I'm trying to hit a DC X and I get an extra +1 to my roll (effectively making the DC X-1). This works out to be the probability of hitting X-1 exactly. You can relabel the Y axis P(roll = X-1). (That's why the d20 probabilities are 0.05 from 2 to 21 and not 1 to 20, since if I start at DC 2, I gain 5% success if I get +1, but if I start at DC 1, I gain nothing if I get an extra +1)

I'm graphing the same thing for both evens and odds. The reason there are points at odds on the 2*3d6-10 curve is that I'm really rolling 2*3d6-(d2-1), and so I really can hit both odd and even numbers.



I mean, it depends on how your measure the discrepancy. I'm actually measuring the discrepancy at those values (as simple difference in probability) and including it in the plot. Not sure why you think I haven't taken it into account. If you want to measure discrepancy as a ratio, well then yeah, it's big, since one of the terms is zero. But you've never in this whole massive thread suggested that your problem was in the way I was comparing probabilities.

In your hypo, you make a clear point of scaling both the rolls and the DCs. You do this because it would be immediately obvious you were talking about different systems if you did not. Yet, your entire argument here is tgat you can do this and it's the same.

Again, the argument I have issue with is that d20 does not differ significantly from 3d6, proved by scaling 3d6 by 2 and recentering, but still using the original DC scheme. This is improper. I've tried that argument cleanly, I've tried it by showing the mismatch in range for the cumulatives, and I've tried by pointing out the gaps in the probabilities, all to try show this. So far I've failed, so here's a final go:

Explain how the same set of DCs generate the same results if I use a d20 in one and 2*d20-10 (with or without your kludge as you please). If you cannot, please revisit your argument that the same DCs used for d20 vs 3d6 comparisions are still valid for 2*3d6-10 (kludged as you wish) v d20 comparisons.

 

[Esker]

Have you been under the impression this entire time that I was claiming that 1d20 was similar to 3d6? I've never claimed that. The claim is that 1d20 is similar to 2*3d6-10. Of course 3d6 and 2*3d6-10 yield different results vs the same DCs.

The scaling and shifting were never part of the proof, they are part of the system.

 

[Ovinomancer]

Have you been under the impression this entire time that I was claiming that 1d20 was similar to 3d6? I've never claimed that. The claim is that 1d20 is similar to 2*3d6-10. Of course 3d6 and 2*3d6-10 yield different results vs the same DCs.

But, they are not, because the DC are at different scales in each. DC exist only in steos of 2 in the 2*3d6 scheme, because you cannot roll at half step intervals. That you can imagine, and even compute, a different probabilty question for the half steps doesn't mean you can suddenly roll those numbers. And, as you note in your hypo, rolling non-existent numbers is a bit of a problem. You can't use a resolution mechanic that's at a lower resolution (heh) than your targets.

In other words, while you can imagine and do math to get a probability fir a half step target number, the functional result of this is that the half step doesn't exist -- the probability of rolling above the half step is shared by the probability of rolling the next highest full step or greater. It's not a separate event, it is the same event with a slightly rephrased pribability question.

Example: if I ask what the odds of rolling greater than a 12 on 2*3d6-10, this is tge same event as answered by asking what tge idds of rolling 13 or greater. It's not a separate event -- it's tge same exact thing. Yet, you've asked these two as if they are the sane and plotted then in the sane PDF as if they are the same. You've completely missed this, even when your PDF no longer sums to 1.

Half steps don't exist in the scaled 3d6 scheme, just like fractional steps don't exist in the unscaled 3d6 scheme.

 

[Esker]

You can't use a resolution mechanic that's at a lower resolution (heh) than your targets.

I mean, sure you can, you just lose some distinctions between DCs. But that's what the confirmation mechanic corrects for. Here's another simplified hypothetical to zoom in on a particular aspect of the situation.

Suppose I don't have any d20s on hand, and so for the night I decide I'm going to roll a d10 and double the result. First: do you agree that if the target number I need is odd, then this produces the same results as rolling a d20? If I would have needed a 19 or 20, now I need a 10 (which becomes 20). If I would have needed a 17,18,19 or 20, now I need a 9 or 10 (which become 18 or 20).

So far so good?

Now, on even DCs my success chances are too high. If I would have needed a 20, which should have a 5% chance, now I will get a 20 with a 10% chance. So, to correct for that, I roll a d6 whenever I tie the DC, and subtract 1 on a 3 or lower. Now what are my chances of getting that 20 I need? Well, the only way to do it is to roll a 10 on the d10, and a 4 or higher on the d6. That's a 10% * 50% = 5% chance. What are the chances of hitting a target of 18 or better? I can either: roll a 10 on the d10 (ignoring the d6), which happens 10% of the time, or I can roll a 9 on the d10 and a 4 or higher on the d6, which happens 10% * 50% = 5% of the time. In total, 15%. Just as on a d20.

So, if I lost all of my d20s and used this system instead, would it affect the game at all?

Example: if I ask what the odds of rolling greater than a 12 on 2*3d6-10, this is tge same event as answered by asking what tge idds of rolling 13 or greater. It's not a separate event -- it's tge same exact thing.

I agree. And?

Yet, you've asked these two as if they are the sane and plotted then in the sane PDF as if they are the same. You've completely missed this, even when your PDF no longer sums to 1.

I explained in my last post what I was plotting. It actually does give you the same values as the PMF, just shifted by 1. Did you add up the values? They actually do sum to 1.

 

[Esker]

I should add: the last red and blue graph only corresponds to the shifted-by-one PMF if you roll the d2 and subtract on a 1 after every roll, not just when you tie the target. But even if you don't do that it's still valid for what it was constructed to be: the increase in success chance if you gain an extra +1 on your roll.

 

[FormerlyHemlock]

It turns out that, in a roll over/under system, replacing 1d20 with 3d6 doesn't do much of what people want.

Mostly what it actually does is double bonuses and DCs, outside of crit hit/miss zones.

snip

D&D is a roll over/under system, not a roll exactly. Looking at "roll exactly" curves is looking at the derivative, when you should be looking at the values -- which is the cumulative distribution function. On anydice, this is "roll at least" or "roll at most" graphs.

Note that 2d10 is about half way, and represents about x1.5 modifiers.

Hmmm. Two thoughts:

(1) Generally the complaints people have about d20 are precisely about those crit hit/miss zones, especially something that should require a crit success for a beginner but is straightforward for an expert, like navigating rapids or climbing a reverse-slope cliff.

(2) Dungeon Fantasy/GURPS gets extra mileage out of the 3d6 bell curve by utilizing margin of success (either directly or via Quick Contests, e.g. for a feint, or resisting a spell). Since the bulk of the probability curve will always be around 9-12, each extra +1 not only boosts your success rate but boosts your average success margin. I don't know whether or not that makes GURPS not strictly "roll over/under" by your definitions, but it does suggest that merely shifting D&D to a 3d6 bell curve without utilizing margins of success will not yield the same benefits you get from Dungeon Fantasy/GURPS.

Thank you especially for insight #2.