In Dimension 20, when there's a particularly important roll to be made, they bring out the Box of Doom, which is ultimately just a box with a big skull-mountain thing built into it (I believe a camera is lodged in the skull to look down at the results). During these rolls, DM/GM Brennan Lee Mulligan will typically explain precisely what is required for a success or failure - "You've rolled a 23 on your Athletics check to shove Sky Captain Vaughn off of the deck of the Blood Raptor, potentially sending this notorious pirate, who slew your father and bombarded your home, off into the great blue sky to his death. He has a +12 to acrobatics, meaning that he needs to roll an 11 or higher to avoid being shoved."
The roll is made, suspenseful music plays, and there's either a cheer as the villain is sent to his demise, or a gritting of teeth and a new strategy to be formed if the bad guy avoids it.
What's notable here, though, is that the probability is all out there in the open. The players know that there's a 50% chance that something good happens. Box of Doom rolls also often happen when something very bad might happen - Vaughn and the player character might have been in opposite positions, and the player might be rolling not to be tossed off the ship.
D&D is built on a somewhat hierarchical assumption: the Dungeon Master is the master of the game's world (traditionally, some deep dark dungeon, hence the name). And one of the ways the game is built gives the DM the privilege of knowledge the players don't have. There are, of course, secret traps, hidden doors, monsters hiding around corners or a gelatinous cube hidden plain sight on that far-cleaner-than-it-should-be hallway.
There are a lot of features that allow you to add bonuses to certain rolls, but you're forced to commit to using these features before you know if they'll be helpful. Bardic Inspiration, for example, lets you make your d20 test initially, and lets you see what you rolled, but you need to make the decision to add your extra die before you know whether you have a success. Sure, sometimes it's obvious - if you got a 25 on your attack roll, you probably won't use Bardic Inspiration, and if you got a total of 3 on a Perception check, that d6 is unlikely to change the result.
But there can be this guessing game, sometimes, about whether something is worth using. For example, the Great Weapon Master feat allows you to take a -5 penalty to your attack roll for a +10 bonus to damage. This tends to give a net bonus to your damage output even at fairly high Armor Classes, but naturally, using this on a Zombie is going to be far more reliable than using it on a heavily-armored creature like an Iron Golem. Part of the feat's power is the ability to choose when to use it, but knowing that can sometimes push players into meta knowledge, guessing at a monster's AC (is that big earth elemental really hard to hit because it's made of rock, or really easy to hit because it's so slow?)
So... what if we were just open?
With DCs for ability checks and saving throws, what if we just came out and said "the dragon breathes, and I need you all to make a DC 15 Dexterity saving throw?" or "You might recognize the deity depicted in the fresco with a DC 18 Religion check."
To be sure, this will empower players. Features like Bardic Inspiration or an Artificer's Flash of Genius (especially the latter, which gives a flat bonus) will become extremely reliable. But I think of this a bit like the Shield spell - while I've had a DM try to change its functionality mid-campaign (she relented) you get to know if Shield is going to do its job or not before you commit to casting it. The DM calls out the total of their attack, and if that meets your AC but doesn't exceed it by five or more, you can cast that spell and negate the effect.
I think it can feel tempting as a DM to want to lure your party into wasting resources like this - I've had times when a powerful monster has an effect with a saving throw that's over twenty prompt a Flash of Genius that only raises a character's result to a 19. Or, conversely, sometimes the DC is not that high, and when someone rolls a 16, they think, "I'm level 15, that monster must have a higher DC," only to realize that the DC is only 14, and that they didn't need to spend that resource.
But... that's not very fun, is it?
Given how powerful player characters get, especially at high levels, DMs might feel tempted to try to squeeze out as much of an advantage as they can get. But I think that transparency in this regard could lead to more satisfying gameplay - and it might also give players a sense of whether they're taking a big risk trying to pickpocket that archmage or not.
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