d20 vs 2d20

[jmucchiello]

Let me start by saying there are no right answers to this question.

In most d20 games, there are two circumstances for d20 rolls: rolls vs DC and opposed rolls. But math-wise, this means that some rolls have linear probability distribution with 20 discrete results and some rolls have triangular distribution with 39 discrete results where central results are more frequent. Is there a reason combat and saving throws are always linear rolls while SOME skill rolls are linear and SOME are triangular? Why is Stealth sometimes done against passive perception and sometimes done against an active perception roll?

Should there be two different random distributions in the same game system? And why not use the triangular one in more places? Why is AC fixed at 10+? Why not roll it? Why are saving throw DCs fixed? why not roll it for each spell? In both cases, the normal roll is linear but it could be triangular. We don't do it, my guess, because it would be too many die rolls: Announce casting a spell, roll d20, calculate DC, target rolls d20 to meet the target. Fixed DCs skip the roll and calculate steps.

And if triangular rolls are "bad" for combat and saving throws, why are they appropriate for SOME skill checks? What is special about skill checks? Is it because there are no auto failure/auto success in skill checks? Why is that so important that the range of the roll needs to expand from 20 to 39 result states with a different probability distribution?

That's a lot of questions but at the root the question is why are there two different probability distributions used in a system where the rules say there is one roll mechanic: roll d20, add mods, check if you reach the DC.

It's easy to "fix". Just use passive skill values for all skill check DCs. If the player is being stealthy, they roll against the guard's passive perception. If the player is being sneaked up on, they make a perception check against the sneak's passive stealth value. (Let the player make the rolls that affect their characters, right?)

But ultimately, does it even need "fixing"? I don't know. It was just something I noticed and it's been bugging me. I just want to hear some folks plea the case of having the two different die rolls. Maybe there's something I'm overlooking.

 

[Harzel]

I think I am missing something - where does the 39 come from? If you roll d20 vs. d20, there are 400 possible outcomes. Are you thinking of the sum of the dice? I don't think the sum is relevant to two d20s rolled as a contest.

 

[Dausuul]

I think I am missing something - where does the 39 come from? If you roll d20 vs. d20, there are 400 possible outcomes. Are you thinking of the sum of the dice? I don't think the sum is relevant to two d20s rolled as a contest.

The sum is not, but the difference is. Rolling a 15 versus a 12 has the same result as rolling a 7 versus a 4.

 

[ccs]

But ultimately, does it even need "fixing"? I don't know. It was just something I noticed and it's been bugging me. I just want to hear some folks plea the case of having the two different die rolls. Maybe there's something I'm overlooking.

Yes, there's something really important that you're overlooking:
The fact that most of the people who've ever played, & will ever play, this game DON"T CARE about this "problem" you perceive. For them, rolling 2d20 to get the same result (success/failure) 1d20 would yield would be pointless.
Rolling 1d20 might not be the perfect answer math-wise, but it's a nice simple solution that works for the majority.
So no, it's not worth fixing.

 

[The Crimson Binome]

In most d20 games, there are two circumstances for d20 rolls: rolls vs DC and opposed rolls. But math-wise, this means that some rolls have linear probability distribution with 20 discrete results and some rolls have triangular distribution with 39 discrete results where central results are more frequent.

The distribution doesn't really matter, though, does it? If you're rolling a d20 to hit DC 11, or if you're rolling a d20 in order to beat another d20 roll, you're still as likely to succeed or fail. Rolling a 1 when someone else rolls a 20 is not inherently worse than rolling a 10 against a static DC 11.

Rolling the extra die does provide some normalization, so if one side has a larger bonus, then that side is even more likely to win an opposed contest than they are to win against a static DC. The magnitude of that shift should also scale with the initial disparity, too; if you're far superior to your opponent, then you benefit even more from rolling two dice. If normal combat switched to a two-die model, then it would be even easier for the players, since PCs are already at a statistical advantage against anyone they go against.

Honestly, though, I'm not sure if any of that really matters. The basic conceit of the d20 system puts playability ahead of everything else. This is a game where blindness has the same mechanical effect on swinging a sword as being poisoned or falling prone does. It should be overwhelmingly obvious that all such decisions were made in order to keep the game moving more quickly. So the probabilities are not as accurate as they could be? It's not like the alternative is all that much more realistic or verisimilitudinous. You're still accounting for the basic relevant factors, and it's equally fair for all parties, so that's the important thing.

 

[Shiroiken]

To answer the original question, it's done this way for speed of play. AC and save DCs being static numbers is basically assuming they always use passive results, because it would double every roll (attack and defense for each attack, or DC and save). Same for Passive Perception against Stealth.

Oddly, I have considered the impact of removing static target numbers. It would create a more chaotic environment, assuming the DM hides their rolls (most do), because you wouldn't know if they have a high AC/Save, or if they just rolled really well. If I were to incorporate this idea, I would want to do so with more dice involved, such as using the Skill die variant (plus possibly using something like that for ability modifiers too).

 

[CapnZapp]

What is the question?

 

[Li Shenron]

The distribution doesn't really matter, though, does it? If you're rolling a d20 to hit DC 11, or if you're rolling a d20 in order to beat another d20 roll, you're still as likely to succeed or fail. Rolling a 1 when someone else rolls a 20 is not inherently worse than rolling a 10 against a static DC 11.

I think the distribution does matter vs static DC exactly because it changes the probability of success and failure, if we roll 2d10 instead of 1d20, the probability of success vs high DC or failure vs low DC will be significantly smaller.

OTOH I don't actually think that the triangular distribution of the difference in opposed rolls matters, because for those rolls by default it's not the difference that matters, but just who scores higher.

A more statistics-savvy person could do a full math analysys and give a definite answer.

 

[dregntael]

I have wondered the same thing before so I did a quick calculation. I calculated the following situations:

• Situation 1: attacker rolls with 1d20+A against passive defence of 10+B, attacker wins on a tie
• Situation 2: attacker rolls with 1d20+A against active defence of 1d20+B, attacker wins on a tie

The results actually surprised me: in situation 1, the chance of success is 0.55 + (A-B)/20 (so the attacker has an advantage of 5%), while in situation 2 the chance of success is 0.525 + (A-B)/20 (so a 2.5% advantage for the attacker). So apart from a very small difference in the constant term, the probabilities are exactly the same! So you can probably safely switch between passive and active defence without affecting the balance at all.

Here are my calculations for those who want to check them:[sblock]Let N := A-B

P(1d20+A >= 10+B)
= P(1d20-10+N >= 0)
= 1/20 * SUM_{k=1..20}(P(k-10+N >= 0))
= (11+N)/20
= 0.55 + N/20

P(1d20+A >= 1d20+B)
= P(1d20-1d20+N >= 0)
= 1/400 * SUM_{k=1..20}(SUM_{l=1..20}(P(k-l+N >= 0)))
= 1/400 * SUM_{k=1..20}(k+N)
= 1/400 * (SUM_{k=1..20}(k) + 20*N)
= 1/400 * (210 + 20*N)
= 0.525 + N/20
[/sblock]

EDIT: I made a mistake in the calculation of situation 2, the right probability is 0.525 + 0.04875 * (A-B) - 0.00125 * (A-B)^2 if A >= B and 0.525 + 0.05125 * (A-B) + 0.00125 * (A-B)^2 if A < B. Here is a plot of the difference using wolfram alpha. You can see that the attacker should prefer a passive defence unless their bonus is a lot lower than the defender's.

[sblock]
P(1d20+A >= 1d20+B)
= P(1d20-1d20+N >= 0)
= 1/400 * SUM_{k=1..20}(SUM_{l=1..20}(P(k-l+N >= 0)))
= 1/400 * SUM_{k=1..20}(max(0,min(20,k+N)))

if A >= B (N >= 0):
[...]
= 1/400 * SUM_{k=1..20}(min(20,k+N))
= 1/400 * (SUM_{k=1..(20-N)}(min(20,k+N)) + SUM_{k=(21-N)..20}(min(20,k+N)))
= 1/400 * (SUM_{k=1..(20-N)}(k+N) + SUM_{k=(21-N)..20}(20))
= 1/400 * ((20-N)(21-N)/2 + (20-N)*N + 20*N)
= 1/400 * (210 + (39/2)*N + -(N^2)/2)
= 0.525 + 0.04875 * N - 0.00125 * N^2

if A < B (N < 0):
[...]
= 1/400 * SUM_{k=1..20}(max(0,k+N))
= 1/400 * (SUM_{k=1..(-N)}(max(0,k+N)) + SUM_{k=(-N)..20}(max(0,k+N)))
= 1/400 * (SUM_{k=1..(-N)}(0) + SUM_{k=(-N)..20}(k+N))
= 1/400 * SUM_{k'=0..(20+N)}(k')
= 1/400 * (20+N)(21+N)/2
= 0.525 + 0.05125 * N + 0.00125 * N^2
[/sblock]

 

[5ekyu]

Let me start by saying there are no right answers to this question.

In most d20 games, there are two circumstances for d20 rolls: rolls vs DC and opposed rolls. But math-wise, this means that some rolls have linear probability distribution with 20 discrete results and some rolls have triangular distribution with 39 discrete results where central results are more frequent. Is there a reason combat and saving throws are always linear rolls while SOME skill rolls are linear and SOME are triangular? Why is Stealth sometimes done against passive perception and sometimes done against an active perception roll?

Should there be two different random distributions in the same game system? And why not use the triangular one in more places? Why is AC fixed at 10+? Why not roll it? Why are saving throw DCs fixed? why not roll it for each spell? In both cases, the normal roll is linear but it could be triangular. We don't do it, my guess, because it would be too many die rolls: Announce casting a spell, roll d20, calculate DC, target rolls d20 to meet the target. Fixed DCs skip the roll and calculate steps.

And if triangular rolls are "bad" for combat and saving throws, why are they appropriate for SOME skill checks? What is special about skill checks? Is it because there are no auto failure/auto success in skill checks? Why is that so important that the range of the roll needs to expand from 20 to 39 result states with a different probability distribution?

That's a lot of questions but at the root the question is why are there two different probability distributions used in a system where the rules say there is one roll mechanic: roll d20, add mods, check if you reach the DC.

It's easy to "fix". Just use passive skill values for all skill check DCs. If the player is being stealthy, they roll against the guard's passive perception. If the player is being sneaked up on, they make a perception check against the sneak's passive stealth value. (Let the player make the rolls that affect their characters, right?)

But ultimately, does it even need "fixing"? I don't know. It was just something I noticed and it's been bugging me. I just want to hear some folks plea the case of having the two different die rolls. Maybe there's something I'm overlooking.

Grapple checks are d20vd20 and are combat checks.

That said, in my game, i dont do opposed rolls. The players roll vs static DC scores.

The only case where opposed rolls would occur would be if it were pc on pc for my games and thats rare.