Dice math... help? Advantage vs bonuses

I'm bad at this,  could someone show me the equations for figuring out the probabilities between advantage/disadvatage compared with numerical bonuses?  I've seen people post similar stuff in the past but can't seem to find the thread.


Is advantage better/worse/equal to disadvantage? And, If I need a set number (or higher) to hit on a d20, at what point do numerical bonuses/penalties outweigh having advantage and disadvantage dice?
Run a search, this thread has come up many, many times.

In short, it really depends on the value at which the attack hits.  If you hit only on a 20, advantage more-or-less doubles your chance to hit.  If you miss only on a 1, it increases your chance to hit only by a relatively small amount, but decreases your chance to miss by a relatively large amount.  Etc.
Harrying your Prey, the Easy Way: A Hunter's Handbook - the first of what will hopefully be many CharOp efforts on my part. The Blinker - teleport everywhere. An Eladrin Knight/Eldritch Knight. CB != rules source.
Advantage is better than Disadvantage. (logical, considering the definitions of the words...)

Advantage anydice.com/program/1c5d
Disadvantage anydice.com/program/1c5e

Nutshell: With Advantage you have about a 50% chance of rolling a 15 or higher on your d20; with Disadvantage you have about a 50% chance of rolling 7 or lower.

Supporting an edition you like does not make you an edition warrior. Demanding that everybody else support your edition makes you an edition warrior.

Why do I like 13th Age? Because I like D&D: http://magbonch.wordpress.com/2013/10/16/first-impressions-13th-age/

AzoriusGuildmage- "I think that you simply spent so long playing it, especially in your formative years with the hobby, that you've long since rationalized or houseruled away its oddities, and set it in your mind as the standard for what is and isn't reasonable in an rpg."

Run a search, this thread has come up many, many times.

In short, it really depends on the value at which the attack hits.  If you hit only on a 20, advantage more-or-less doubles your chance to hit.  If you miss only on a 1, it increases your chance to hit only by a relatively small amount, but decreases your chance to miss by a relatively large amount.  Etc.



Yeah I know it has, I just suck at searching.  Maybe I should have made this a "how the hell do I actually find what I want?" thread lol.  When I search I get tons of topics not related to the question


I know advantage is better than disadvantage... I just mean as far as probabilities go.  Is giving 1 person an advantage roll equal to instead giving his opponent a disadvantage role?    
The rightmost columns here is what you're looking for (from the middle 2d20 column to the right):



As you see, if the DC of your check is 11, rolling 2d20 take higher gives  you the equivalent of a +5 bonus on the check.

At other values the equivalence is not so precise, but you can see that for a roll targeting 19, you get almost the same as a +2.

For the rest, check the colors. Green means you have a better change of reaching that result with the flat bonus, red means 2d20 has a better chance.

So unless the roll targets 1-2 or 19-20... rolling 2d20 is better than a flat +2 bonus, for example.



You have to consider, too, that a +2 or +5 bonus can get you as high as 22 or 25 on the roll. 2d20 will never get you there no matter how lucky you are with the dice.


Hope this helps.

More info, if you want, on this quite nice post:
critical-hits.com/2012/06/11/dd-advantag...
Thank you so much Rasta
(I still dislike this mechanic. A lot. Almost enough to not play DDN because of it.)

Supporting an edition you like does not make you an edition warrior. Demanding that everybody else support your edition makes you an edition warrior.

Why do I like 13th Age? Because I like D&D: http://magbonch.wordpress.com/2013/10/16/first-impressions-13th-age/

AzoriusGuildmage- "I think that you simply spent so long playing it, especially in your formative years with the hobby, that you've long since rationalized or houseruled away its oddities, and set it in your mind as the standard for what is and isn't reasonable in an rpg."

(I still dislike this mechanic. A lot. Almost enough to not play DDN because of it.)

It's nice find agreement on the fundamentals.  Still, as divisive as the whole thing is, they can solve all of it with a simple module:

ALTERNATIVE ADVANTAGE: When you have advantage on a check, you gain +2. When you have disadvantage on a check, you suffer -2.

Of course, then we need to trust them to not tie too many things into the second die roll - if there's an ability that says, "when you have advantage, the second die is guaranteed to be at least a 10," or, "when both dice come up less than 7," then it loses transparency.
The metagame is not the game.
(I still dislike this mechanic. A lot. Almost enough to not play DDN because of it.)




Why?  I understand it skews things and makes the game different from a numbers perspective.  But other than that, doesn't it simplify things?  Rather than adding this and that and subracting this and that,  just roll 2d20 and add your static bonuses.  Doesn't it speed up game play?
Why?  I understand it skews things and makes the game different from a numbers perspective.  But other than that, doesn't it simplify things?  Rather than adding this and that and subracting this and that,  just roll 2d20 and add your static bonuses.  Doesn't it speed up game play?

I guess it depends on how much you care about modeling a system in detail (circumstance X should grant a greater bonus than circumstance Y, and the the flat bonus shouldn't change based on what the DC is), against just speeding up gameplay.

I mean, the system is plenty fast enough already. I'd prefer a little bit of extra room to play with in modeling, rather than every situation coming down to just advantage/neutral/disadvantage. It feels like an unnatural constraint in system design.

The metagame is not the game.
(I still dislike this mechanic. A lot. Almost enough to not play DDN because of it.)



ALTERNATIVE ADVANTAGE: When you have advantage on a check, you gain +2. When you have disadvantage on a check, you suffer -2.

Of course, then we need to trust them to not tie too many things into the second die roll - if there's an ability that says, "when you have advantage, the second die is guaranteed to be at least a 10," or, "when both dice come up less than 7," then it loses transparency.



First (just to be clear) I like advantage.


That said - although I agree with this in theory, advantage is worth far more than a mere +2 (and thus is harder to get than was combat advantage in earler editions).  For the range most DCs are in (7 to 15), advantage is closer to a +4 or +5.  A simple rule ought to reflect that. 


Thus a simple rule would be:

ALTERNATIVE ADVANTAGE:  When you have advantage on a check, you gain +4. When you have disadvantage on a check, you suffer -4.

But this changes how it works with regards to high DCs (advantage makes you more likely to be able to do things you could have done without advantage - it does not make you able to do things you could not have done otherwise).   Aside:  This is one of the things I really like about it.


Thus a less simple rule would be:

ALTERNATIVE ADVANTAGE:   When you have advantage on a check, you increase the result of the die roll by 4 (before adding any other modifiers to the roll).  The value of the die roll cannot be increased above 20.


But I'd rather just use advantage.


Carl  

    
ALTERNATIVE ADVANTAGE:  When you have advantage on a check, you gain +4. When you have disadvantage on a check, you suffer -4.

But this changes how it works with regards to high DCs (advantage makes you more likely to be able to do things you could have done without advantage - it does not make you able to do things you could not have done otherwise).   Aside:  This is one of the things I really like about it.

One of the important things about such a broad alternate rule should be that it doesn't upset game balance too much, which is why I'd be fine with keeping it at 2 rather than 4 - one of the more consistent arguments from the anti-2d20 camp is that rolling two dice is too much of an advantage, so bringing that down to 2 means it will be less of a big deal and it would be less likely to push your result up over 20.

From a purely mechanical standpoint, your "increase the result of the die roll, but not over 20" version would be a much better analogue to existing advantage, but I'm not really comfortable with that level of complexity in such a basic rule.  (Not that it's objectively difficult, but only that it's more difficult for a group which is fine with just using a flat bonus anyway.)
The metagame is not the game.
I knew I was forgetting something major, and it finally came back to me.

The biggest reason I dislike 2d20 is because it's hard to intuitively quantify.  Advantage is always better, and disadvantage is always worse, but how much you gain or suffer will vary depending on your base chance of success.  Increasing my objective success rate by 10% means something to me (although not nearly as much as increasing my objective success rate by 20%). 
The metagame is not the game.
The math posted earlier is wrong. From here www.anydice.com. You can see that a normal roll hitting 5, 10, 15, or 20 are as follows:

5 = 80% chance of success.
10 = 55%
15 = 30%
20 = 5%

Advantage does this to the numbers:

5 = 96%
10 = 79.75%
15 = 51%
20 = 9.75%

So the equivalent bonuses you end up with are (5% per point rounded to closest point):

5 = +3
10 = +5
15 = +4
20 = +2

So as you can see WotC really doesn't understand the math involved. The closer you are to needing to roll an 11 on the dice, before bonuses and penalties, the more advantage and disadvantage matter. The further away the less it matters.

Which math are you referring to?
Because that is literally the same math that Rasta posted earlier.

(And 20 by your scale would be +1, not +2)

The whole point of the article he linked to was that the value of advantage varies based on the target roll (not the DC, the target roll that would result in the DC post modifiers).

It's a fairly good visual for quickly seeing the ranges where advantage compares to falt bonuses.

And yes, you both make essentially the same point.  Advantage matters more when the target roll is closer to 11.
Advantage matters more when the target roll is closer to 11.


As often as this statement is made, I continue to find it highly suspect.

It is only true in the sense that advantage equates (roughly) to a larger flat bonus when the target roll is closer to 11.  However, I have seen no justification for objectively stating that "a +1 bonus is a +1 bonus is a +1 bonus".  The actual impact of a flat bonus is certainly context-dependent, requiring you to consider both the existing probability of success/failure and the meaning of said success/failure.  Since a flat bonus is not a fixed metric against which to measure another bonus, you cannot simply say "Advantage matters more when the target roll is closer to 11."

(Please note the following lose their validity at the extremes of automatic failure or success.)
 
Example: In terms of a proportional increase in expected damage, advantage matters more the lower your current probability of hitting.

Example: In terms of a proportional decrease in the probability of failure, advantage matters more the higher your current probability of success.

Actually, both of these examples are also true for flat bonuses, though not in precisely the same fashion.   
Advantage matters more when the target roll is closer to 11.


As often as this statement is made, I continue to find it highly suspect.

It is only true in the sense that advantage equates (roughly) to a larger flat bonus when the target roll is closer to 11.  However, I have seen no justification for objectively stating that "a +1 bonus is a +1 bonus is a +1 bonus".  The actual impact of a flat bonus is certainly context-dependent, requiring you to consider both the existing probability of success/failure and the meaning of said success/failure.  Since a flat bonus is not a fixed metric against which to measure another bonus, you cannot simply say "Advantage matters more when the target roll is closer to 11."

(Please note the following lose their validity at the extremes of automatic failure or success.)
 
Example: In terms of a proportional increase in expected damage, advantage matters more the lower your current probability of hitting.

Example: In terms of a proportional decrease in the probability of failure, advantage matters more the higher your current probability of success.

Actually, both of these examples are also true for flat bonuses, though not in precisely the same fashion.   




Well, yes, by changing the metric from "flat percentage increase in success" to "proportional increase in success" you have also changed the meaning of "matters" in that context.

However, the comparison being made is in the difference between advantage and a flat bonus.  Even in those circumstances, the difference in proportional increase of success is still greater for advantage (as compared to a flat bonus) at middling target rolls than at extremes.

Example: (Proportional increases in parenthesis {rounded})
                        base   Adv.                +2
Target roll   3:    90    99(10)          100(11)
Target roll   4:    85    97.75(15)     95(12)
Target roll   7:    70    91(30)          80(14)
Target roll 11:    50    75(50)          60(20)
Target roll 15:    30    51(70)          40(33)
Target roll 18:    15    27.75(85)     25(66)
Target roll 19:    10    19(90)          20(100)

Notice how the differences between the proportional increase of advantage and the proportional increase of +2 are greater in the middle numbers and less at the extremes?

This is what we mean when we say that advantage "matters more" for middling target rolls.
This is the statement I responded to:
 
Advantage matters more when the target roll is closer to 11.


Now, perhaps you mean something by this other than the way I interpret it, but I'm quite certain that what people generally mean by this is the following:

When the target number is 11, the change in the probability of success from advantage is the same as that of a +5 flat bonus.  As you move away from 11, the equivalent flat bonus decreases until it is approximately +1 at the ends (before reaching automatic success/failure).  Thus, advantage has its greatest effect in the middle.

So, if that's not what you're saying, then I will say that I interpreted your statement in a different manner than what was intended.  However, I'm also saying that without serious qualification the explanation in the previous paragraph is wrong.  It's akin to a situation in which you measure two objects with a ruler, finding that object A measures 2 inches and object B measures 3 inches, and so you conclude that object B is longer than object A.  The problem is that the ruler being used changes the length of an inch depending on where it is and which way it's pointing.  So, actually, it's pretty useless for telling you which object is longer.

A thought experiment: Suppose I have the power to increase the probability that your hair is about to catch on fire by 1%.  That's a flat bonus.  However, it means something completely different if the probability is currently 0.0000001% as compared to 95%.  In the former case it's the 1% increase that's going to make you nervous.  In the latter case, it isn't the 1% increase that's freaking you out.  And that's the point: a flat bonus is not a static metric against which to measure the effect of a bonus.

As a further thought, suppose advantage is equivalent to a +2 bonus in one situation, and is also found to be equivalent to a +2 bonus in a second situation.  That means advantage is worth the same as some other bonus that gives you +2 in these situations, but it does not necessarily mean that advantage is of equivalent benefit from one situation to the next, since there's no guarantee that you could say that about the +2 bonus either.  Similarly then, if you find that advantage is equivalent to a +1 bonus in one situation, and equivalent to a +2 bonus in another situation, it does not necessarily follow that advantage is "more useful" in the second situation, because you cannot necessarily determine which flat bonus is "more useful" within its given context.

(Edit: found a mistake in this paragraph, in which I should have said "success" instead of "failure."  Fixed.) 
If you were to say "the proportional increase in the probability of success from advantage is greater when your probability of success is lower," a similar notion to the first example in my previous post, then it means as your probability of success decreases, the proportional increase in your probability of success from advantage increases.  Your own chart illustrates as such.

If you want to say "the difference between the proportional increase in success from advantage and the proportional increase in success from a particular bonus is largest around a target number of 11" then fine, though I don't see any particular qualification in your previous post to suggest that's what you mean.  (I further assume that you mean ratio rather than difference, since on your chart the difference is greater at 15 than at 11.)  Even so, it might be true for particular bonuses, not all flat bonuses in general.  Were you to add another column to your chart for a +7 bonus, it would look as follows:
                       +7
Target roll  3:  100(11)
Target roll  4:  100(17)
Target roll  7:  100(43)
Target roll 11:  85(70)
Target roll 15:  65(86)
Target roll 18:  50(233)
Target roll 19:  45(350)

In this case the difference (and ratio) is clearly not biggest around 11.  So even that interpretation requires you to specify which flat bonuses you are referring to.

So: I find the statement "Advantage matters more when the target roll is closer to 11" to be highly suspect.  At the very least it requires significant qualification to be true, and without such is likely to be inaccurate and misleading.

Edit: In a further effort to understand if we are just talking about different things, I wonder if the intention of your statement could be rephrased as "if you compare the benefit of advantage to a particular flat bonus, then depending on the flat bonus you may find advantage has a greater impact than said bonus over an interval of target values, and if so this interval will centre around 11."  If that's the case then I will agree that it's true, but I don't immediately see why it's useful or follows from "Advantage matters more when the target roll is closer to 11."  To me it makes more sense to interpret that as "Advantage matters more when the target roll is closer to 11 (than it does when you are further away from 11," similar to how I would interpret, say, "How I perform on stage matters more when I have an audience in front of me" as to mean it matters more than when I don't have an audience in front of me.

So as you can see WotC really doesn't understand the math involved. The closer you are to needing to roll an 11 on the dice, before bonuses and penalties, the more advantage and disadvantage matter. The further away the less it matters.



You are mistaken.  This tells us that WotC knows exactly what they are doing with advantage/disadvantage.

If a task is average difficulty, (i.e. you need a 10-12 to do it), then Advantage makes it a near certainty.  You have advangate, you do really well.

However, if what you are trying to do is really difficult (DC of 18 or higher) then having advantage is only a small boon, and in some cases will still be impossible unless you have trained the correct skill, or have a 20 in your relevant stat.
A further illustration:

Suppose we are defining a mechanic for death saves, and selecting between two options.

Option A:  Each round a player makes a d20 check (DC 11).  After 4 failures the player dies.
Option B:  Each round a player makes a d20 check (DC 6).  After 2 failures the player dies.

Suppose also that we are deciding between them on the basis that advantage may exist for these checks, and we want to minimize the significance of advantage.  Which one should we choose?  You might say B, because both involve a d20 check, but option A's check is at 11, where advantage equates to the larger flat bonus of +5.

Let's first note that for each option the expected number of checks until death is 8.  (And no, I'm not saying that makes the distributions identical, or that one objectively favours players more than the other without consideration of context.  I'm saying they are comparable in the sense that they have a quality--expected number of rounds until death--that can be used as a point of comparison, that in this case that quality begins the same, and that we can thus analyze the impact of advantage on this basis.)

Option A with advantage:  Your probability of failure on the d20 check goes from 0.5 to 0.25.  The expected number of rounds until death doubles from 8 to 16.
Option B with advantage:  Your probability of failure on the d20 check goes from 0.25 to 0.0625.  The expected number of rounds until death quadruples from 8 to 32.

Well, yes, by changing the metric from "flat percentage increase in success" to "proportional increase in success" you have also changed the meaning of "matters" in that context.


It isn't that I'm "changing" the metric.  It's that I'm bothering to (a) specify the metric, and (b) consider whether that metric "matters" in terms of measurable in-game consequence.

Edit: I'd have to think on this some more, but my initial impression is that it is very defensible to say that the significance of advantage is actually more stable across the contexts of various target numbers than a particular flat bonus, rather than less so. 
A further illustration:

Suppose we are defining a mechanic for death saves, and selecting between two options.

Option A:  Each round a player makes a d20 check (DC 11).  After 4 failures the player dies.
Option B:  Each round a player makes a d20 check (DC 6).  After 2 failures the player dies.

Suppose also that we are deciding between them on the basis that advantage may exist for these checks, and we want to minimize the significance of advantage.  Which one should we choose?  You might say B, because both involve a d20 check, but option A's check is at 11, where advantage equates to the larger flat bonus of +5.

Let's first note that for each option the expected number of checks until death is 8.  (And no, I'm not saying that makes the distributions identical, or that one objectively favours players more than the other without consideration of context.  I'm saying they are comparable in the sense that they have a quality--expected number of rounds until death--that can be used as a point of comparison, that in this case that quality begins the same, and that we can thus analyze the impact of advantage on this basis.)

Option A with advantage:  Your probability of failure on the d20 check goes from 0.5 to 0.25.  The expected number of rounds until death doubles from 8 to 16.
Option B with advantage:  Your probability of failure on the d20 check goes from 0.25 to 0.0625.  The expected number of rounds until death quadruples from 8 to 32.

Well, yes, by changing the metric from "flat percentage increase in success" to "proportional increase in success" you have also changed the meaning of "matters" in that context.


It isn't that I'm "changing" the metric.  It's that I'm bothering to (a) specify the metric, and (b) consider whether that metric "matters" in terms of measurable in-game consequence.

Edit: I'd have to think on this some more, but my initial impression is that it is very defensible to say that the significance of advantage is actually more stable across the contexts of various target numbers than a particular flat bonus, rather than less so. 

I feel like you're being intentionally nit-picky here.

Throughout the thread (and every other thread that discusses advantage) the comparison has been in the flat percentage shift in probability for advantaged accross target rolls, versus the flat percentage increase in probablility for a flat numerical bonus to the roll.

What you're discussing is the the game impact of ANY bonus and how it effects character action.

The problem with that comparison is that you have no baseline to compare it to.  Yes, advantage by your defenition "matters" more at high TN's, because ANY bonus matters more at high TN's.

The only way to understand the impact of the advantage mechanic is to compare it to the only other mechanic that represents what advantage is meant to achieve. (ie the flat bonus ).   The whole point of comparison is to guage which mechanic has a greater impact at various TN's.  And the only meaningful way to do that is to compare the flat percentage in probability increase.

You examples above only compare advantage with itself.  Those results are not useful in a comparison of where advantage matters because there is no baseline.


Don't get me wrong here.  Your evaluatiopn of the in game effects of advantage have a lot of merit.  You just seem to be ignoring that the rest of the thread has been discussin advantage as compared to a flat bonus.  That is the implied portion of my statement which spawned you to counter a point I didn't make.

No one is arguing that having a bonus to your roll matters more for difficult rolls.  The point of confusion with advantage is that it's effective bonus is a bell curve based on the target roll.   

That is the only point people have been attempting to clarify. And again, that is what I mean when I say that advantage "matters more" an TN 11.  It means that for that one specific roll (not the in game effects of that roll, but the roll itself) the difference in pass/fail probability is more greatly effected at TN 11 than at the extremes.
Thank you JihVed. That is exactly the issue I was trying to describe in this thread community.wizards.com/go/thread/view/758... regarding skill training granting advantage instead of the skill die. 
Wasn't really able to respond for a few days, but I do wish to do so.

As far as I'm concerned, I'm not being nit-picky (deliberately or not) or ignoring what has been said in the thread.  I actually appreciate that the OP asked specifically when advantage is more powerful than particular flat bonuses rather than a more vague description.  That isn't the point though.  The relevant portions of the post that spawned yours, and of your post that I quoted, are:
The closer you are to needing to roll an 11 on the dice, before bonuses and penalties, the more advantage and disadvantage matter. The further away the less it matters.


And yes, you both make essentially the same point.  Advantage matters more when the target roll is closer to 11.


I don't see how you can take the first of those to be anything other than a comparison of advantage with itself across two different contexts; it's a question of the change in significance of advantage rather than a question of magnitude.  Further, it seems to be based entirely on saying that advantage equates to a larger flat bonus in one context than another, and rather than leaving it at that, making a logical leap that advantage is more significant in the first context using the equivalent flat bonuses as a measuring device.  I acknowledged that that might not have been what you meant (the point you didn't make).  It might not be the point meant by any particular quote; I really don't know.  But I think it is the most reasonable interpretation of that particular language, and it has appeared in almost exactly that form enough times over the forums for me to think it is meant precisely that way quite frequently.  If the language of the thread really was restricted to measuring the equivalent flat bonus for any particular target number then fair enough, but I don't believe the language of the thread (along with many others) is so restricted.

I also don't see any particular reason that it shouldn't be discussed in such a fashion.  Why can you only talk about the significance of a bonus relative to flat bonuses?  Why is that the only way to understand the impact of the advantage mechanic?  I would presume that we similarly want to consider the significance of using flat bonuses for certain situational benefits if they are in use, rather than leaving it as the end of the discussion.  I'm not breaking entirely new ground here; the very first response in this thread made a distinction between relative changes in probability of both success and failure, and people have otherwise talked about how much is gained with the implicit indication that you can determine significance more or less directly from a flat bonus.  You can make the comparison for the purpose of determining which mechanic (advantage vs. particular flat bonus) is more powerful at various target numbers, but I see no reason to stop the discussion there, and I'm quite sure that others haven't.

I will also say that I am certainly not suggesting that proportional decrease in the chance of failure is "the" or "my" definition of significance, though I think in many situations it's a very good one.  However, even taking that as the measure of significance of a bonus for a moment, it is not true that any bonus is more significant at higher probabilities of success than lower; you could certainly construct modifiers to a probability distribution such that it is not the case.

Edited slightly for clarity.
The answer to your question is that mainly everything you describe can be equally applied to every other mechanic and set of rolls in the game. Advantage/disadvantage doesn't alter the results you are talking about compared to regular rolls. Yes it has an effect, but that same effect happens when you use a normal roll across multiple rolls.


I assume that the question(s) to which you are referring is/are "Why can you only talk about the significance of a bonus relative to flat bonuses?  Why is that the only way to understand the impact of the advantage mechanic?"

I never said that advantage and a flat bonus are categorically different in this regard; indeed I'm pointing out the exact opposite, that advantage and a flat bonus are categorically similar here.  Once you move into language comparing "significance" or how much something "matters" across different contexts, using flat bonuses as a point of comparison is largely pointless since there is no basis for thinking flat bonuses provide a stable measurement of such things, and taking flat bonuses to be so leads to some pretty questionable conclusions.  However, saying that they are in the same category does not mean that the discussion ends; that would be like saying all increasing functions are essentially the same and you can't analyze them any further.  Even if you were to determine that two bonuses behave in a qualitatively similar fashion that doesn't mean you can't consider quantitative differences. 

For example, if you compare probability of success with or without advantage, or with or without a +2 bonus, you will find that as your target number decreases (and your base chance of success goes up) the ratio of probability of failure with the bonus to probability of failure without the bonus gets smaller and smaller in each case.  In other words, from a certain (perfectly reasonable) point of view, the significance of both bonuses is larger as you change the context to one of a larger base probability of success.  However they certainly don't do so in the same way.

I'm not sure if every current mechanic behaves qualitatively similar; in order to check I'd have to look at every current mechanic.  However, it is certainly possible to construct dice mechanics that do not.  It's a bit tricky to go in this direction; it's generally much harder to determine a suitable dice mechanic from a desired behaviour than to determine behaviour from a mechanic, but here's one example:

I'll call it the "1 in 4" bonus.  When you make a d20 check with this bonus, you also roll a d4.  If the d20 check passes, you succeed.  If the d4 results in 1, you succeed.  This is behaving in a categorically different manner: the ratio of probability of failure with the bonus to the probability of failure without the bonus is always 0.75, except where it is undefined for no probability of failure.