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The math in D&D 5e is pretty solid with a few rough spots, and most of your skill rolls should be DC 13.
That's a great starting point when we assume that:
Skill checks are only made in dangerous or uncertain situations.
The average adventurer is reasonably competent at core adventuring tasks.
Superfluous (non-impactful) d20 rolls are elided.
5e is built around point buy, with the assumption that your primary ability score will have a +3 bonus with a 16 or 17 base. (Clerics have +3 Wisdom, Rogues have +3 Dexterity, Barbarians have +3 Strength, etc.)
That means at level 1, your attack rolls have a +5 bonus and your save DCs are 13. This is important but we'll get back to it later.
Obviously, the first three things are a matter of GM skill, and the fourth is a matter of the GM's table, but let's work off these (safe) assumptions.
Skill Ratings
In terms of skills, you have several "ratings" that I'll describe:
Superb (+7). You have a natural talent (+3 ability score) and expertise (+4).
Excellent (+5). You have a natural talent (+3 ability score) and proficiency (+2).
Good (+3). You have a natural talent (+3 ability score) and no proficiency (+0), or you have an middling talent (+1 ability score) and proficiency (+2).
Average (+1). You have a middling talent (+1 ability score) and no proficiency, or low talent (-1) and proficiency (+2).
Poor (-1). You're bad at the thing (-1 ability score) and no proficiency.
Now, there are all kinds of permutations of these ratings, but 5e math is built around point buying your "key ability score" up to 16-17 (+3 bonus) at level 1.
You can have slight variations to these numbers, but the most important ratings are the high and low ends and how they relate to DC 13.
Superb (+7) = 75% success.
Excellent (+5) = 65% success.
Good (+3) = 55% success.
Average (+1) = 45% success.
Poor (-1) = 35% success.
These numbers are all quite sensible, I'd say; if your rogue has invested expertise in lockpicking, there's no reason he shouldn't be hitting the mark three-quarters of the time. (If you don't like these numbers, you can tweak the DCs by 1-3 points.) There's plenty of opportunities for failures in a dungeon!
Now, as soon your proficiency bonus increases, the math immediately begins to fray into ruin because the proficiency bonus gets bigger and non-proficient characters get left behind as the rest of the game scales forward.
Critiques of Bounded Accuracy
People in the Online 5e Community often complain about these numbers and how they relate to bounded accuracy, most of them directed at one of the following:
The d20 is too swingy! I need a bigger number to obviate rolling the die.
My skill bonus isn't enough higher, and it should be for [reasons].
I can't reliably hit a DC 30 ability check.
These are all poorly-reasoned.
Firstly, the d20's perceived "swinginess" is known as variance. Yes, you have an equal chance of rolling a 1 or a 20 on the d20, but it doesn't matter because the absolute % of success or failure matters. If you are rolling 3d6 vs. DC 11, you have a 50% chance of success with or without the vaunted bell curve. This is identical to rolling 1d20 vs. DC 11! (3d6 vs. DC 13 is almost identical to 1d20 vs. DC 16, 3d6 vs. DC 15 is almost identical to 1d20 vs. DC 19, and so on.)
Secondly, why do you need a larger bonus to your ability checks? I've just explained why DC 13 is great. The only time you need a larger bonus is if the GM is inflating the DCs beyond the suggested 10 (Easy), 15 (Medium), or 20 (Hard).* You only need larger skill bonuses if the GM is regularly calling for DC 21+ ability checks - and, frankly, if the GM is grossly miscalibrating the DCs, he's simply going to inflate them upward to counter your increased skill bonus. (This is an unsavory business and should be censured.)
Thirdly, you're not supposed to reliably hit a DC 30 ability check. It's nearly impossible, and it should be incredibly rare. High-level adventurers with expertise can hit DC 30s; level 1 fresh-out-of-fighter-college adventurers cannot.
*Notice that DC 13 falls between 10 and 15 and is the starting save DC for spellcasters.
How 5e Fails at Bounded Accuracy
As I mentioned previously, the ability check math begins to fall apart at higher levels because of how proficiency bonus scaling works.
Superb (+17). Your ability score grows to +5, and the expertise bonus grows to +12.
Excellent (+11). Your ability score grows to +5, and the expertise bonus grows to +12.
Good (+5 to +7). Your ability score grows to +5, and you don't have proficiency, or your ability score remains at +0 to +1 and your proficiency bonus grows to +6.
Average (+1). Your ability score rises to +2 via half-feats, and you're not proficient.
Poor (-1). Your ability score stays at -1, and you're not proficient.
I'll provide a chart at key breakpoints. Remember that levels 4, 8, and 12 are ability score increases or feats. (I've skipped the feats at level 16/20 as I don't consider them mattering overmuch. Adventurers might boost one of their ability scores by +1 here or there but the damage is already done via scaling.)
Looking at these numbers, you can see the issue: I've established why DC 13 is a sensible ability check DC, and yet it eventually becomes meaningless to some characters while remaining challenging for others.
How do we fix this? Let's analyze the issue:
Proficiency bonus increases over time.
Proficiency bonus is an on/off button.
Expertise multiplies proficiency bonus.
We can tackle any of these, but let's start with #1. Guess what? We can't. D&D is predicated upon Number Go Up, for good and ill, so removing that aspect of gameplay is a no-go.
Let's move on to #2, the proficiency binary. By making proficiency bonus a yes/no button that scales upward, you're making the "yeses" stronger and comparatively making the "nos" weaker. You could fix this in various ways (perhaps adopting a more incremental proficiency system not unlike Pathfinder 2e's), but I suggest the following:
All characters improve their proficiency bonus over time.
A powerful wizard, despite his frailty, is still better at adventuring tasks like climbing and swimming by virtue of having gone on a lot of adventures.
And #3. I hate expertise because it is a boring mechanic that directly flies in the face of bounded accuracy. It's terrible and must needs be excised from the game. Here's the simplest fix:
Expertise provides a one-time bonus to ability checks.
Let's consider how this could be implemented without grossly disrupting 5e's underlying math.
Proficiency Bonus, Revised
Here's my suggested house rules:
Reduce proficiency bonus by 2 across the board (+0 at level 1).
If you have proficiency with a skill, you add +2 to your ability checks.
If you have expertise with a skill, you add +4 to your ability checks.
Why these numbers? Here's how things look.
As you can see, the Superb and Excellent characters stay well ahead of the game, and the Average and Poor characters get some small benefit. One would hope that this would incentivize them to pick up a few ability score boosts to shore up their weaknesses as they go along.
Further Musings
With the way proficiency bonus works, the saving throw math in 5e begins to collapse at higher levels for all the same reasons. You should probably just give everyone proficiency in saves.