Game designers obsess over probabilistic precision when designing their TTRPG mechanics. They'll run formulas through AnyDice and pore over the exact results, tweaking convoluted dice mechanics and numerics to achieve the perfect spread of results.
Then they totally disregard how those mechanics play at the table.
TTRPG mechanics need to "feel good" in play.
You cannot design a good RPG around mathematically-perfect probabilities because RPG is not a math game. An RPG is a social game that usually involves math. As a social game, it leans heavily into human psychology to function--and human psychology is unquantifiable and, moreover, fickle. The human mind is not a logical reasoning machine; it is emotional and volatile in spite of its capacity to reason.
An RPG designer must be cognizant of the lower-level processes of the human brain, what you might describe as "vibes" or "feels," and how they impact play at the table.
Should you disregard this, you will fail.
An Example: Advantage & Disadvantage
One of the resounding successes of D&D 5e is the mainstreaming of "roll twice and keep the higher/lower die" in the form of advantage or disadvantage. Why? Because it's a mechanic that works with human psychology. The exact percentages of advantage/disadvantage do not matter. In fact, they're downright sloppy.
Rolling 1d20+0 vs. a DC 11 in D&D 5e has a 50% chance of success. With advantage, this is pushed to a 75% success rate. Pushing the DC to 16 reduces the initial success rate to 25%. With advantage, the success is bolstered to 43.75% success.
In one circumstance, advantage adds a hearty 25% to the base success rate, and in another, it adds a more meager 18.75%.
This is less mathematically "perfect" than adding a +5 bonus to the die roll, but it doesn't matter because the human brain is bad at math. A +5 has less of a psychological impact that rolling 2d20 and keeping the highest die. In short, advantage feels better than a +5.
Feels Bad, Man
A mechanic that "feels bad" (or even "feels wrong") when used is a mechanic that should be revised because it leaves a negative psychological impact on the participants and thus worsens the experience. (Note that this does not mean that every "feels bad" moment should be removed from the game, but if your mechanics consistently lead to "feels bad" or "feels wrong" moments, you should take stock of the situation and consider redesigning them.)
As another D&D-related example: people consistently complain about the d20 being "swingy" in D&D. Why? Not because of any mathematical reasons--it's not probability, it's that rolling a d20 feels like a gamble unless you've got huge bonus to the roll. The d20 vs. DC mechanic is a fast, simple, easily-understood mechanic, but it often produces outcomes that "feel bad" to a number of players. The fact that you have a 5% chance of a nat 1 on a roll even if you're an expert at the task leaves a bad taste in many players' mouths.
The Subjectivity of "Feels Good" vs. "Feels Bad"
Different players have different playstyles, and some players will play with a mechanic and deem it as "feels good," while others will play with it and deem it as "feels bad." A TTRPG must know its target audience and devise mechanics accordingly. In short, just because some players think a mechanic "feels bad" doesn't mean that you must redesign it--unless it's a significant proportion of your target audience!
Example #1: Pathfinder 2e
Pathfinder 2e relies heavily on min/maxing small bonuses and penalties for its system to function. Your goal, as a player, is to stack up the +1s and +2 for your team while imposing -1s and -2s on your enemies so that you can score critical hits and cause your enemies to suffer critical failures. These bonuses are often applied on-the-fly and require recalculation--something I hated from the days of D&D 3e.
Couple this with an intensive character generation minigame and my eyes glaze over. Everything about this is a "feels bad" for me--but Pathfinder 2e players love this kind of min/maxing. Pathfinder 2e is designed for them, and the mechanics "feel good" to them.
Example #2: Fate
I detest the way that Fate handles Aspects. For those not in the know, Aspects are words / phrases / descriptors placed upon a person, place, or thing that describe something about it that is important to the story.
In a traditional RPG, if the players enter a darkened warehouse, the GM might impose a penalty on attack rolls or a bonus to sneaking around unseen, but this isn't the case with Fate. Until an Aspect is written down, it's not important enough to matter. This means that if the player characters enter a darkened warehouse, that darkness has no impact on the game until someone writes that the area is Black As Midnight or Veiled in Shadows or Obscured by Darkness. Even then, the Aspect has no true mechanical weight--that is, it has no impact on the dice rolling part of the game--unless you either spend metacurrency or you take a specific action to add stacks of free metacurrency to it.
This feels utterly wrong and backwards for me, but Fate players love the mechanic because it's not about there being darkness present, it's about the darkness being important to the narrative: if nobody makes darkness an Aspect, it's not important to the story.
TL;DR Wrap-Up
The long and short of this rambling essay:
Your game is not a probability sim. Math matters less than you think.
Consider redesigning mechanics that "feel bad" in play.
Know your target audience and design TTRPG mechanics around their tastes and preferences.
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