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

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?

# 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?

## -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%.

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