Statistically, which is more of a severe penalty on a d20 roll, Disadvantage or a -10 penalty?

Question

As a DM, I sometimes rule that a player can roll at Disadvantage or with a -10 penalty on a d20 roll. However, I've not been particularly consistent with this as I don't know what is a bigger hindrance.

Are the chances of success higher when rolling the d20 with Disadvantage or with a -10 penalty?

Answer 1

TL;DR Statistically a -10 is worse for the roller than disadvantage -- except for attack rolls

We'll start by ignoring automatic successes on a natural 20. Suppose you need to roll some number S to succeed. If you roll with disadvantage, your probability of success is

$$\left(\frac{21-S}{20}\right)^2$$

However, if you roll with a -10 penalty your probability of success is 0 for any S greater than 10 and otherwise $$\frac{10-S}{20}$$ A little bit of math shows that the probability of success is always higher when you roll with disadvantage. I've graphed out the probability of success below.

Attack rolls change this a bit

Attack rolls always succeed on a natural 20 and always fail on a natural 1.

You only have a 1/400 chance of rolling a 20 with disadvantage, but the odds are still 1/20 when you roll with a -10 penalty. This changes the probability of success when you take a -10 penalty to 1/20 when S is greater than 10. Again, a little bit of math shows that if you need a 17 or higher, then the -10 penalty has a higher probability of success than rolling with disadvantage.

You have a 39/400 of rolling a 1 with disadvantage but the odds are still 1/20 when you roll with a -10 penalty. However, this only comes into pay when S is less than -8, and how often does that happen?

Answer 2

-10 is too much

Imposing a -10 penalty is the same as increasing DC by 10. This is way too big in terms of 5e bounded accuracy. It basically increases a task difficulty from "Medium" to "Very hard".

A practical example:

For low-level adventures, typical static DCs for "average difficulty" tasks are usually around 12; if you're proficient or have a 14 in the stat, you succeed 55% of the time. Expertise, or proficiency + 14 stat is a 65% success rate. Disadvantage reduces them to 30.25% and 42.25% respectively. A -10 would reduce those to 5% and 15%.

(kudos to @ShadowRanger)

Although the rules suggest using penalty instead of disadvantage for passive rolls, it is just 5:

If the character has advantage on the check, add 5. For disadvantage, subtract 5.

This is not exact equivalent for very low or very high DCs. For instance, for DC 16 with zero modifiers introducing a -5 penalty makes this roll impossible.

I'd advise against using such penalties

You said that

I sometimes rule that a player can roll at Disadvantage or with a -10 penalty on a d20 roll

I suggest to reconsider this approach:

• It adds unnecessary cognitive load for players, slowing them down.
• You can just set a reasonable DC instead of imposing penalties.
• It's also faster to just roll two d20s and choose the lower.

Also keep in mind that penalties stack, while (dis)advantages do not:

If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage

The 5e rules prefer disadvantage to penalties for the sake of simplicity. The idea is to let players focus on what's important (the game itself) while skipping unnecessary details (what kind of hindrance to choose).