8
\$\begingroup\$

I know the critical hit probability for normal critical range (natural 20) is:

  • normal: 0.05
  • advantage: 0.0975
  • disadvantage: 0.0025

But I don't know how to actually calculate this, and consequently, I don't know how to calculate these probabilities for an expanded critical range (e.g. 19,20 or 18,19,20).

So:
what are the actual formulas to calculate the probability for a critical hit, with an expanded critical range, and advantage/disadvantage?

\$\endgroup\$

1 Answer 1

15
\$\begingroup\$

For expanded critical range, count the number of rolls that give a crit, and divide by 20.

  • So if you crit on 19 or 20, that's \$\frac{2}{20} = 0.1\$.

  • If you crit on 18-20, that's \$\frac{3}{20} = 0.15\$.

  • If you crit on all the numbers from \$x\$ to 20, then your probability on a straight roll is \$\frac{21-x}{20}\$

Now if you have the probability for a normal roll, you can figure out the probabilities with advantage/disadvantage like this:

  • If you have disadvantage you can only crit if you roll a crit on both dice. The probability of this is the same as multiplying your original chance to crit by itself. So if you crit on 19-20, you get \$0.1 \times 0.1 = 0.01\$.
  • If you have advantage, you crit if either dice rolls a crit. You might think this just double your crit chance, but because some rolls involve rolling a crit on both dice, those scenarios would be counted twice, so we have to subtract the case where both dice crit. So again if you crit on 19-20, then your crit chance with advantage would be \$2\times0.1 - 0.1 \times 0.1 = 0.2 - 0.01 = 0.19\$.

TLDR: you can use these formulae.

  • Straight roll: \$p = \frac{21-x}{20}\$ where \$x\$ is the lowest value you crit with.
  • Disadvantage: \$p^2\$
  • Advantage: \$2p - p^2\$
\$\endgroup\$
4
  • \$\begingroup\$ for a 'critical mis', I simply need to reverse the formulas? \$\endgroup\$
    – Jacco
    Oct 26, 2023 at 19:13
  • 9
    \$\begingroup\$ Another way of computing the odds with advantage is \$1-(1-p)^2\$ (which expands out to your formula). You can interpret that the odds of critting are the odds you do not fail to crit with both dice (with fun double negatives). \$\endgroup\$
    – Blckknght
    Oct 26, 2023 at 19:33
  • \$\begingroup\$ If you use Black Cat DM's Familiar, for a character sheet, you can export an expected damage analysis which handles this, it shows you the expected damage for different attack modes accounting for the chance of a critical hit, with advantage, with neither advantage/disadvantage and with disadvantage. \$\endgroup\$
    – Dughall
    Oct 27, 2023 at 10:42
  • 1
    \$\begingroup\$ This is a great answer, but in the explanation of the simple probability I think “count the number of rolls” would be clearer than “add up”. The example makes it clear but I think that language might trip up some folks not used to maths. \$\endgroup\$ Oct 27, 2023 at 20:41

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .