How do I calculate the probability for a critical hit, with an expanded critical range, and advantage/disadvantage?

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

• normal: 0.05

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?

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\$$
• for a 'critical mis', I simply need to reverse the formulas? Oct 26, 2023 at 19:13
• 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). Oct 26, 2023 at 19:33
• 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. Oct 27, 2023 at 10:42
• 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. Oct 27, 2023 at 20:41