Analytics Check

2021/02/01

In our previous post, we dug into the mathematics of DnD 5e’s advantage and disadvantage mechanic. Looking at the probability expressions we derived allowed us to understand their general behavior. Here we go through a few helpful rules of thumb to provide more practical results to apply directly in your games.

A few tips

As advantage and disadvantage only applies to d20 rolls in DnD, these results come from our probability expressions with $n=20$.

• The single most likely outcome is a natural 20, occurring in about 1-in-10 rolls.
• The least likely outcome is a natural 1, occurring in 1-in-400 rolls.
• There is about a 1-in-2 chance of rolling a 15 or higher.
• There is about a 9-in-10 chance of rolling a 7 or higher.
• Rolling a natural 20 is more likely than rolling below a 7.

• The single most likely outcome is a natural 1, occurring in about 1-in-10 rolls.
• The least likely outcome is a natural 20, occurring in 1-in-400 rolls.
• There is about a 1-in-2 chance of rolling a 6 or lower.
• There is about a 9-in-10 chance of rolling a 14 or lower.
• Rolling a natural 1 is more likely than rolling above a 14.

Why are these true statements?

A fairly straightforward way to show that all of the above statements are true is to examine the PMF and CDF graphs that we made for d20 rolls at the end of the last post. We show them again below:

By mousing over or tapping on the graphs you can see the exact probabilities of different situations. Mousing over the first graph shows that the probability of a natural 20 with advantage is 0.0975, which is about 1-in-10 and is higher than the probability of rolling a 6 or lower with advantage as seen in the second graph (which is only 0.09). By playing around with these, you can come up with all sorts of useful results!

Here’s a concrete DnD 5e example — a Champion Fighter scores a critical hit when rolling a 19 or higher from its Improved Critical feature. Looking at the graph above, we see that our fighter with advantage will crit on average in 1 out of 5 hits! With an extra attack and Action Surge at their disposal, an advantaged fighter is force to reckoned with.