One D&D's UA 2022: Character Origins has introduced a concept of gaining inspiration without the DM awarding it. This is one of the biggest changes in One D&D so far, and I want to determine the impact it will have on balance and gameplay. Inspiration used to be pure DM fiat and now it is partially mechanical, so this is a huge change.
I want to know how often the players will earn Inspiration without the DM granting it using the new Inspiration rules compared to the original 5e rules.
All we really know is that with the new d20 test, a critical results in the gaining of an inspiration point. Additionally, there are new race/background mechanics that provide clear paths to inspiration. But how often you roll and whether you are at advantage or disadvantage will depend entirely upon your table.
Whether or not it's more or less than what your own table does now is going to be determined by your table. There will be an increase in mechanically provided inspiration, but whether or not that changes the rate of inspiration at your own table is going to be variable.
At my own table, I know that remembering to give inspiration is hard - and remembering to use it even harder! By adding a mechanic to 'help' a DM 'remember' to give it out, it could increase the rate at tables where the DM isn't on top of it as much (but not necessarily help in players using it!)
My only note is that in my experience Inspiration leads to a lot of fun. People take chances when they have advantage, and a lot of fun stories are all about when things go big (or go wrong in a big way.)
Iron Kingdoms uses an inspiration mechanic that also refreshes quite often to encourage bigger moves and I found it to work really well (and conversely, withholding that mechanic made everyone more wary when the GM did that.) If D&D is moving towards this ideal, then having more opportunities to get and use inspiration will assist in this.
We can first estimate the number of Insprations from natural 20s alone. The relevant rule from the new Character Origins resource is:
A player character also gains Inspiration when rolling the 20, thanks to the remarkable success.
Since the probability of rolling a 20 is 0.05, we can handwave the answer here as something like "approximately one twentieth of the d20 tests in a session will yield Inspiration". This is handwaving because the frequency of d20 tests is highly dependent on playstyle. I've played for DMs who rarely call for skill checks, at tables that are very light on combat, but also for DMs who call for all the skill checks and run multiple fast combats per session. These things make the frequency of d20 tests vary wildly table to table.
While we cannot talk about a "typical D&D table", as there is no such thing, Crit Role Stats collected highly detailed statistics on Critical Role's Mighty Nein campaign. Using this data, we can at least pin down how often the Mighty Nein would have earned inspiration via rolling natural 20s during their time on Exandria.
During their 141 session campaign, totaling 483 hours of actual play time, the Critical Role cast rolled 604 natural 20s, which comes out to:
Ergo, if the frequency of d20 tests at your table, was similar to Matthew Mercer's table, those are the numbers you can expect using the new Inspiration rule from One D&D.
Adding to this, the Character Origins pdf provides two other methods that do not rely on the DM. First, any human characters automatically gain Inspiration at the end of a long rest. And, once again, the frequency of long rests per session is going to be highly dependent on playstyle and campaign. I could not find any statistics on the number of long rests taken by the Mighty Nein, so I cannot provide a data-based estimate for you.
The other method provided is through the 1st level Musician feat:
As you finish a Short Rest or a Long Rest, you can play a song on a Musical Instrument with which you have Tool Proficiency and give Inspiration* to allies who hear the song. The number of allies you can affect in this way equals your Proficiency Bonus.
Again, a feature dependent on frequency of rests.
If you have humans and musicians in your party, you can essentially assume that the entire party will start with Inspiration at the end of every short or long rest. Without humans and musicians, it just depends on how often you roll the dice, but once every 75 minutes is at least an accurate number for an n=1 study of a single play group.
The probability of gaining Inspiration on a saving throw, skill check or attack roll heavily depends on how much often one of these rolls is called at your table. Down below I enlist some possible styles: these are not the sole DnD styles that one can play, but they are examples for illustrating how even a hard-coded mechanic could heavily depend on the playstyle.
In some campaigns fights are more predominant with respect to other aspects of the game (e.g., roleplay, using particular skills in social environment), a lot of attack rolls are made, hence the probability to get inspiration grows.
Let's take the guidelines from the DMG and suppose that in an adventuring day 8 fight encounters occur; on average, a fight lasts 3 rounds. Consider a 4th level Fighter, which will use its Action Surge trait: this means that they will get 25 (\$=3\cdot 8 +1\$) attack rolls (no opportunity attack considered here). The probability to get at least one inspiration in a day it is around 72.26% (computed as 1 minus the probability to get no inspiration at all: \$1-\left(19/20\right)^{19}\$). This computation does not take into account that the inspiration can be used for an attack roll. Moreover, since this is a hack&slash adventure, maybe the number of fights is greater than 8, or the fights last for more than 3 rounds.
Even a wizard that cast a lot of spells that require attack rolls may get inspiration in this type of game (provided they have enough slots): indeed, even if it seems that they can not crit, a 20 on the attack roll still grants inspiration.
Pay attention that the above probability refers to an adventuring day, not to a game session: and adventuring day may span several sessions.
An adventure may involve more skill checks rather than attack rolls. I usually play in campaigns which are a good melange of roleplay, combat and social interactions, hence I can not provide any direct experience of this type of game. Anyway, one could imagine a campaign based on political-social interaction and/or spying: in this case, the DM may call several skill checks. In this case providing a statistic is quite hard, just remember the formula $$ P(\text{get at least 1 inspiration on }n \text{ skill checks}) = 1 - \left(\frac{19}{20}\right)^n. $$
Some tables play a DnD game that is pure roleplay: convincing the innkeeper to be poor for a discount on the ale or bribing the city guards for obtaining the permission to enter the walls do not rely anymore on Deception or Persuasion checks (or any other one that the DM decide is appropriate), but only on the acting ability of the player. In this case, getting inspiration via attack rolls, saving throws or skill checks could be very hard.
Rolling the so called d20test is not the only way to mechanically get inspiration: the Resourceful trait of the Human race allows to obtain an inspiration once a Long Rest has successfully ended. The Musician feat allow to play a song in order to give inspiration at the end of a short or of a long rest. The gritty realism variant depicted in the DMG (pag 267) says:
This variant uses a short rest of 8 hours and a long rest of 7 days. This puts the brakes on the campaign, requiring the players to carefully judge the benefits and drawbacks of combat. Characters can't afford to engage in too many battles in a row. and all adventuring requires careful planning.
Besides the possible side effect to engage less fights (i.e., less attack rolls), the new Human race will get an inspiration only after a week and not at the beginning of every day within this playstyle variant. Instead, a character with the musician feat is not affected by the gritty realism variant.
