To what extent is a lower DC skill check with disadvantage harder than a higher skill check without?

Question

I'm wondering how mechanically I could calculate a lower DC challenge with disadvantage as a middle-ground between a higher skill DC and that same DC with disadvantage. I'd like to describe a situation where, for example:

• Playing the harmonica is hard (let's say DC13)

• Playing the harmonica while either riding a unicycle or juggling is
  harder (Maybe DC10 with disadvantage?)

• Playing the harmonica while both riding a unicycle AND juggling is
  really hard (DC13 with disadvantage)

My gut says that DC10 with disadvantage should rank in difficulty somewhere between DC13 vanilla and DC13 with disadvantage, but I have a feeling there's more to it than that.

Answer 1

I agree with your self-answer; in your case, DC changes alone are probably better than disadvantage (and definitely better than DC changes plus disadvantage). Now for some theory...

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DC and skill bonus do not matter; their difference does

Passing a DC 13 skill check using a +3 bonus is mechanically identical to passing a DC 25 skill check using a +15 bonus. The same is true when comparing different DCs with disadvantage. Only two things matter when considering skill checks:

1. Do you have advantage/disadvantage?
2. What number do you have to roll on the d20 to succeed?

Using my previous example:

• In order to pass a DC 13 skill check using a +3 bonus, you need to roll a 10 on the d20.
• In order to pass a DC 25 skill check using a +15 bonus, you need to roll a 10 on the d20.

These two checks are mechanically identical, as are all checks where you have to roll the same number on the d20 (with the same presence/absence of advantage/disadvantage).

How does disadvantage affect difficulty?

Disadvantage makes checks more difficult if those checks are possible and failable.

A check is possible if you have to roll a 20 or less on the d20 to succeed. A DC 20 Intelligence (Arcana) check is not possible for a barbarian with a -1 Intelligence modifier (and no proficiency in Arcana). A natural 20 has no effect on a skill check.

A check is failable if you have to roll a 2 or greater on the d20 to succeed. A DC 14 Charisma (Persuasion) check is not failable for a level 9 bard with a +5 Charisma modifier and Expertise in Persuasion (granting a +13 bonus). A natural 1 has no effect on a skill check.

Any time a skill bonus increases by one, one DC becomes possible, and one DC becomes unfailable. The rogue's Reliable Talent works by redefining failable as: if you have to roll an 11 or greater on the d20 to succeed. The bard and warlock spell glibness raises that to 16 or greater for Charisma checks. Both features narrow the effective range of disadvantage.

Disadvantage never changes a check's possible or failable status. Giving the barbarian advantage on the above Arcana check doesn't suddenly make the impossible, possible. Likewise, giving the bard disadvantage doesn't suddenly make the unfailable, failable.

How does lowering the DC affect difficulty?

Lowering the DC may make impossible checks possible and failable checks unfailable.

By lowering the DC, you are changing the difference between the DC and the skill bonus, and thus you are changing the number needed on the d20 to succeed. The barbarian above is (mathematically) infinitely more likely to pass a DC 19 Intelligence (Arcana) check than a DC 20 one. The impossible has become possible.

Show me the math

Ignoring impossible and unfailable checks, disadvantage affects difficulty non-linearly. Various shortcuts like "advantage is similar to +5/+2.5/etc." can only be useful given a fixed number rolled to succeed.

Ignoring impossible and unfailable checks, lowering the DC affects difficulty linearly. It merely changes the number required to succeed.

Conclusion

Lower the DC if you want more characters to have a chance of succeeding (and specialized characters to possibly never fail). Impose disadvantage if you want to make a task less likely to succeed, while not affecting who could pass or fail. Simultaneously lowering the DC and imposing disadvantage may produce non-intuitive results and should be handled with care.

Answer 2

It would seem I have found my answer, if the math on this linked page is correct. In my example, playing the harmonica at DC13 results in a .400 possibility of success. Add the unicycle (DC10 with disadvantage) and the odds go down to .303, while doing both (DC13 with disadvantage) brings the odds down to .16

Having said that, I now think I was fundamentally misusing disadvantage and DC in my question. The DMG suggests using advantage and disadvantage to reflect either great ideas from the players or changing, temporary circumstances in the scenario.

Advantage and disadvantage are among the most useful tools in your DM's toolbox. They reflect temporary circumstances that might affect the chances of a character succeeding or failing at a task. Advantage is also a great way to reward a player who shows exceptional creativity in play. (DMG, pg 239)

From that standpoint, advantage mechanics aren't well suited to my example, and straight DCs, as suggested by Eternallord66, are probably more appropriate