Trying to calculate "Emphasis roll" on anydice.com

Question

I see two answers on this board for the "Emphasis roll" mechanic and neither of them match how the Emphasis rolls work.

The idea behind the Emphasis roll is that you roll 2 d20's and take the result that is furthest from 10.

However, the probabilities should be arranged against a target number. For example, let's say you were trying to hit a AC 1. You roll 2 d20's and no matter what the rolls and the distance from 10, you should have a 100% chance of hitting a AC 1; none of the solutions posted here represent this fact.

What I'm looking for is a graph that should show what the probabilities are TO HIT an AC of 1, of 2, of 3, of 4 and up using the Emphasis roll mechanic.

The same could be applied for Saving Throws and DC's in D&D for Emphasis rolls.

Answer 1

For reference, the existing questions are here and here.

If you want to know the chance of hitting, you can use the "At Least" button. The default is "Normal", i.e. the chance of rolling exactly a particular number.

Answer 2

One can use the probability distribution function.

The probability distribution function for the Emphasis Roll mechanic is given¹ by $$ P(D) = \frac{2|D-10|-1}{181}. $$

One can use this function to easily compute the probability to get at least a target number on the d20: for example, if the target is 13, one may to compute $$ P(\text{at least 13}) = \sum_{i=1}^{13} P(i) = 53.04\%. $$

Pay attention to mechanics and rules

From the text of the question, it seems that the interest is to apply this to Dungeons and Dragons: pay attention that for attack rolls a 1 on the d20 roll is automatic a miss², regardless of the bonuses.

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¹ See here for the derivation of this formula.

² In the fifth edition, at least, and in most of the editions, to the best of my knowledge.