# Would it be fair to use 1d30 (instead of rolling 2d20 and taking the higher die) for advantage rolls?

Would it be fair to use 1d30 - in place of 2d20 and taking the higher die - for advantage rolls?

Recently I have acquired several d30 dice in a bag of factory seconds. I want to offer my players the option of using a d30 dice instead of using 2d20 dice for Advantage rolls.

The idea is that the d30 is meant to be a high-risk, high-reward option in that you have more chances to roll a high number but, as you are only rolling one dice, there is presumably more risk due to not having an option to take the highest of two rolls.

Whilst I can not find anywhere on the internet that discusses such a situation (most discussions about d30’s boil down to using them for 30-entry tables), I have found probabilities for the dice.

AnyDice is a free website that allows you to calculate the probabilities of rolling any dice and can display the results in various formats. Below are screenshots taken of the probabilities for both 1d30 and 2d20 (take the highest result) to get at least X number:

1d30:

2d20 (taking the highest result):

Based on this data, I want to know if it would be fair to use 1d30 in place of 2d20. By “fair” I mean whether the advantage of the d30 (there being more chances to pass any DC check due to it having higher numbers) would be balanced out by its disadvantage (the lack of an option to take the highest result of two rolls).

The best answers should, using data and examples, explain why using 1d30 instead of 2d20 (take the highest result) is either balanced or unbalanced.

As a number of people have asked what to do about rolls that exceed 20 and critical hits, I feel obliged to provide some clarification:

Firstly, in regards to critical hits, a roll of either 20 or 30 would grant a critical hit. This increases the chance to roll a critical on 1d30 to 0.066, making it so there is more chance to score a critical than on a regular d20 roll, which has 0.05 chance to get a critical hit, but less than 2d20 (taking the highest result) at 0.0975 chance.

Whilst this means you are less likely to score a critical with a d30 as opposed to 2d20 (taking the highest result), you have an extra ten numbers with which you can beat a check with, you have a 36.66% chance to roll 20 or more on a d30.

Secondly, any rolls above 20 on the d30 are equal to 20. This does not mean that all over 20 rolls are critical however, a number that was rounded down to 20 is not a critical hit (unless it was a 30 before rounding it down, in which case the strike is a critical due to initially rolling the 30). This makes it so a character can’t use a d30 to pass a DC30 check without modifiers or any other kind of bonus.

In relation to anything that increases the critical hit range, such as by expanding the range for a critical hit to 18-20, the numbers 18, 19, 20 and 30 would all count as being critical hits in this example. Whilst yes, this would mean you are less likely to score a critical hit with the d30 (13.33% compared to 15% on the d20 if the range was 18-20), this is balanced by having more of a chance to hit, as mentioned earlier.

# It's inherently unbalanced

You're using ways of determining success/failure using methods yielding different probabilities. For this reason it's not fair/balanced.

That being said, let's do a more detailed analysis of the probabilities of achieving a certain result using both methods.

The probability of rolling at least n is 1-(probability of rolling lower with all dice), i.e.

d20 ($$\1\leq n\leq 20\$$ only) $$p = 1 - \left(\frac{n-1}{20}\right)^2$$

d30
$$p = \frac{31-n}{30}$$

The table below shows the probabilities of achieving at least a certain result with both ways of determining the outcome.

$$\begin{array}{r|r|r} min\;roll&d30&max(2d20)\\\hline 1&1.0000&1.0000\\\hline 2&0.9667&0.9975\\\hline 3&0.9333&0.9900\\\hline 4&0.9000&0.9775\\\hline 5&0.8667&0.9600\\\hline 6&0.8333&0.9375\\\hline 7&0.8000&0.9100\\\hline 8&0.7667&0.8775\\\hline 9&0.7333&0.8400\\\hline 10&0.7000&0.7975\\\hline 11&0.6667&0.7500\\\hline 12&0.6333&0.6975\\\hline 13&0.6000&0.6400\\\hline 14&0.5667&0.5775\\\hline 15&0.5333&0.5100\\\hline 16&0.5000&0.4375\\\hline 17&0.4667&0.3600\\\hline 18&0.4333&0.2775\\\hline 19&0.4000&0.1900\\\hline 20&0.3667&0.0975\\\hline 21&0.3333&0.0000\\\hline 22&0.3000&0.0000\\\hline 23&0.2667&0.0000\\\hline 24&0.2333&0.0000\\\hline 25&0.2000&0.0000\\\hline 26&0.1667&0.0000\\\hline 27&0.1333&0.0000\\\hline 28&0.1000&0.0000\\\hline 29&0.0667&0.0000\\\hline 30&0.0333&0.0000 \end{array}$$

Note that only for required rolls of 15 or higher the d30 is actually preferable.

If you're using this for attack rolls, there is an additional drawback of the d30: The probability of rolling a 20 is decreased even compared to rolling a single d20: It's $$\0.0333...\$$ compared to the $$\0.0975\$$ of rolling a d20 with advantage or $$\0.05\$$ for rolling a single d20.

For attacking the chance of hitting is usually 50 to 60% making you hit it 9 to more than 11% less likely to hit the target in addition to decreasing the probability of a critical hit by about 6.4%.

## Conclusion

Using the d30 option is only be prefer preferable for difficult ability checks requiring a roll of 15 or higher.

• Adding something like the plot in this comment would be very helpful here to visually make your point I think. – Rubiksmoose May 15 '19 at 13:37
• Could you explain why you think that methods with different probabilities are inherently unbalanced? It doesn't follow from the rest of the answer. In fact, what I gathered from your analysis is that the d30 is preferable in 50% of the range it covers (numbers of 15+ required) while "2d20, keep highest" is preferable in the other 50%, which seems almost perfectly balanced to me. – Ruther Rendommeleigh May 16 '19 at 11:39
• @RutherRendommeleigh At least in my experience, it's fairly rare to find yourself in a situation where 15+ is required. Generally 8-12 is good enough to get the job done. Assuming this is the normal way of things, 2d20 keep highest will generally be the better way, because there's usually no advantage to rolling far more than what you needed. – CrusaderJ May 16 '19 at 14:54
• @CrusaderJ That seems completely untrue for any type of contest (stealth, grapple, deception). Pretty much all creatures of CR 2 and above have abilities with a DC of 13 at minimum. – Pink Sweetener May 16 '19 at 15:03
• @CrusaderJ I'm inclined to agree. However, this answer starts with, I paraphrase, "it's inherently unbalanced because the two methods don't yield identical results". This part I don't follow. It seems disconnected from the rest of the answer. – Ruther Rendommeleigh May 16 '19 at 15:04

### No, because you're breaking bounded accuracy

Rolling a d30 would let you achieve results far beyond what the character can normally achieve. D&D 5e's 2d20 advantage mechanic specifically prevents this.

D&D 5e follows a design philosophy called bounded accuracy, where the range of numbers a character could roll is intentionally limited. Part of this is the Advantage/Disadvantage mechanic, which elegantly modifies the likelihood of success without changing the range of numbers you can roll.

Compare this to D&D 3.5, where advantageous situations simply gave you a numerical bonus or penalty, and multiple conditions could even stack. A wizard with +5 to Knowledge (Arcana), who can normally not exceed DC 25 check, could acquire numerous small bonuses and make DC 30 check, which is normally beyond their capability.

In short, the problem with using a d30 is the 21-30 range, which makes the impossible possible. D&D 5e's rules assume you aren't supposed to roll that high, and it will create unbalance as a result.

### Also, the d30 is really weak in practice

As per fabian's answer, the odds of rolling under 15 are actually worse with a d30 than with 2d20. It's only statistically beneficial on the hardest 30% of d20 rolls. In fact, it's even rarer than 30%, since AC and save DC are disproportionately likely to cluster around average rolls (e.g. 1d20 plus bonuses against 11 - 17).

This means the d30 is usually weaker than 2d20. Worse, this discrepancy isn't immediately obvious to players, who probably haven't performed a statistical analysis on the rolls, and will be working from too small a sample set of rolls to reliably measure trends. The result is that it's not fair.

But if you do tell them the odds, then the optimum solution is to always choose the d30 when making a known hard roll, and always choose 2d20 against an easy roll. The result is that the d30's weakness can be ignored, so its benefits are not balanced.

• Since your example is of an ability check, do you have thoughts on whether DMG237 (IIRC) which instructs a GM to decide, before asking for a check, whether the thing is even possible mitigates your concern? (I agree with a concern about bounded accuracy, by the way and I think you could make an even more forceful point w.r.t. attack rolls.) – nitsua60 May 15 '19 at 13:43
• @nitsua60 I use skills rather than attack rolls as an example because AC above 20 is really rare in D&D 5e, so it's rare that rolling a natural 29 to hit is any better than rolling a 19. DMG 237 does describe some actions as literally impossible (e.g. shooting the moon). However, skill tasks of DC30 are merely "nearly impossible" (DMG 238), attainable only by extremely high-level characters on very good rolls. Using a d30 makes such tasks possible for characters that previously would have considered them impossible. – Quadratic Wizard May 15 '19 at 14:51
• @NautArch Indeed, though I'm thinking of monster ACs (which rarely exceed 20), since the question mentions offering this ability to players. If monsters also have the ability to roll d30, PCs in full plate have much to fear. – Quadratic Wizard May 15 '19 at 17:48
• Heh, true. But there are a reasonable amount of monsters with greater than 20AC as well. – NautArch May 15 '19 at 17:51
• You could "clip" the d30 so that any roll greater than a 20 is treated as a 20, but that still breaks things: you're four times more likely to roll a "natural" 20 with a d30 than with the higher of 2d20. – Mark May 15 '19 at 23:26

Using the wonderful graph generated by @IlmariKaronen, I want to specifically address one particular point in the question:

The idea is that the d30 is meant to be a high-risk, high-reward option

In a general case (not focusing on any particular ability check value), I would say the risk is shown visually in the graph as the area between the d30 curve and the advantaged d20 curve where the d20 curve is higher. Similarly, the reward is the area between the curves where the d30 curve is higher. I've marked these areas on the graph here:

It's obvious without calculations that the blue reward area is much larger than the red-orange risk area. So, I think "low-risk, high-reward" is a better way to describe the d30 option.

Of course, the other answers make excellent points about bounded accuracy, and that if you know the threshold for the difficulty check you can make easily make the correct choice (with d30 being better for anything >= 15).

We can also look at the average (or expected value). The average of d20 with advantage is about 13.8. The average of d30 is 15.5. So, on average, we can also say that d30 is better than d20 with advantage (just in terms of the resulting value, ignoring the distribution of DC checks a player may face).

(I would also note that, since the die rolls are discrete, area isn't quite the right notion---we'd just want to sum the differences at each point. But it communicates the point effectively, so I'm leaving it as-is.)

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• This is an interesting point, but I'm not sure it really applies. Usually, the only consideration is whether the roll exceeds a threshold or not; the amount by which the roll exceeds the threshold doesn't matter. The area (or sum of differences) would only be applicable in cases where the actual value rolled is relevant. – prl May 15 '19 at 20:33
• I don't think this applies in most cases, due to the fact that rolls that require 21+ pre-modifiers are going to be extremely rare - they would be flat-out impossible in standard 5E, hence probably something you'd only run into when trying to punch way above your weight class. Within the standard range, 1d30 is worse 70% of the time. – Errorsatz May 16 '19 at 0:46
• @prl I think your point is strong when it comes to expected value, which is why I put that near the end with a caveat. For any individual DC, there is a clear choice of which is better---if the player knows the threshold and all that matters is exceeding the threshold, there is no risk/reward trade-off at all---there is one choice that has a higher probability of success. – Gregor Thomas May 16 '19 at 1:20
• In trying to generalize beyond specific thresholds, I do think these concepts/this answer adds value. Adding up the area is adding up probabilities, not values, so I don't think your point about "the amount by which the roll exceeds" applies to that, I agree with @Errorsatz, it would probably be appropriate to truncate the graph at x = 20. Overall, the sum of the differences is 1.675 in favor of 1d30, but if we only consider checks <= 20, then the sum is 0.1583 in favor of d20 with advantage. I'll maybe make some edits tomorrow to include this helpful feedback from both of you, thanks! – Gregor Thomas May 16 '19 at 1:25

# High-Risk, High-Reward is the wrong way of looking at it

You claim that this method is high-risk, high-reward, since it has more variance. Other people have gone into the maths of this more, but I'd like to address this point specifically.

A great example of high-risk high-reward in D&D is a barbarian's Reckless Attack. Attacks against them are more likely to hit for a round - the risk - in exchange for their attacks also being more likely to hit - the reward. You can see how they trade a gain for a greater risk.

Now, the vast majority of d20 rolls in D&D are binary - they succeed or they fail, and they don't much care about the roll apart from that. This means that a choice between two rolling systems, on a given roll, cannot create risk. All they can do is alter the one probability that you care about, and that will either be strictly good or strictly bad, depending on the roll needed.

Thus, choosing what to roll each time does not create the dynamic you want, but rather just bogs down the game with decisions and DC-guessing. You could implement this system on a character level, where each player chooses one option and sticks with it, but this doesn't jive well with high-risk high-reward either - note how Reckless Attack is a choice each turn, because sometimes you don't need the reward, and so you don't want to take the risk. And aside from anything else, having different players using different dice systems is worth avoiding, especially if one comes to be perceived as better and the players using the 'worse' one come to resent this.

• "You could implement this system on a character level, where each player chooses one option and sticks with it, but this doesn't jive well with high-risk high-reward either" - I don't understand this part. I can see how choosing once for a character would be less tactical than choosing once per turn, but how is it less risk/reward oriented? – Ruther Rendommeleigh May 17 '19 at 11:58
• An even better high-risk, high-reward option is the Great Weapon Master feat. You take a -5 penalty to hit in exchange for +10 to damage. – TheCentaur May 17 '19 at 19:58

# It's a bit underwhelming

The updated question states that numbers are clamped to the 1-20 range and details a method for determining criticals that yields a lower chance than the standard 2d20 method. This pushes the d30 option from "situational, but potentially interesting" into "too niche to really be useful" territory.

While it still yields better odds on rolls that require a natural 15 to 20, that advantage alone isn't enough to balance out the increased chance of failure on all other rolls.

As has been pointed out many times before, allowing players to freely choose one of the methods for each roll will (after the players figure out the odds) result in the appropriate method being chosen based on the DC. Due to the d30 option being mostly inferior, a clever player won't pick it in situations where the ability to switch back and forth is limited or the DC is unknown.

Furthermore, the added complexity in determining the result of the roll and/or critical hits (as well as communicating these new rules), while not great, might be enough to dampen players' excitement about rolling an unusual die for some of their checks.