RE: lethality as 1/5 chance of a death:

One of several reasons why my simulation app is still undergoing design revisions is that no one can seem to really agree on what "appropriate" difficulty is in the first place, whether for an encounter or an entire adventure. Less than 20% chance of a permanent death works as a metric, but so do other metrics like "less than 1% chance of TPK", "characters lose at least half their HP and spells," "no single character under this level can successfully complete the adventure," etc. The only rough consensus I've gotten is that appropriate difficulty is, approximately, when the players feel there is some chance of failure, but still usually succeed.

I suppose there's no reason why you couldn't offer a whole bunch of difficulty metrics and just let the user (i.e. the DM) pick which one they want to measure by.

I have done quite a lot of work on games and difficulty, so maybe I can contribute a few concepts.

**Defining Difficulty**

The simplest definition of medium difficulty is that a subject has a 50/50 chance of success at the task.

**Objective versus Subjective difficulty**

Objective difficulty is tricky to pin down. The best concept we've so far come up for it is that a subject exposed to a random task experiences the objective difficulty. After that, practice effect kicks in and we can only measure subjective difficulty. One problem with conceptualising objective difficulty that way is that most subjects evidence a very large performance gain between trial 1 and trial 2, due to simply understanding the concepts of the task for example understanding the control mechanisms. So possibly objective difficulty should be defined as - for a class of tasks, the difficulty experienced on a random member of that class after a formal familiarisation period with the class.

**Performance Limit**

On most tasks, if you smoothly ramp the difficulty manipulation, humans hit a performance limit somewhere (and with fairly good consistency, subject to subject).

**Intermediated Difficulty**

When we contemplate a player controlling an agent (e.g. their character) to do a task, then the difficulty is intermediated through the attributes of that agent. We now need to consider both the leverage the agent has over the task (the maximum performance) and the proficiency of the player in guiding their agent. We can predict that a less proficient player with a more powerful agent might perform as well as a more proficient player with a less powerful agent, but to pin that down to definite probabilities isn't straightforward.

**Ranked Performance**

The most common measures of capability at a game "simplify" the problem by ranking players against one another. Chess Elo ratings are one example. Xbox Trueskill is another. The difficulty of a game task (or scenario forming a set of tasks) can then be said to be the inverse of number of players who can perform that task with a 50/50 chance of success. So if most players can overcome a task or scenario, then it is low difficulty.
Bringing that back to D&D, in sizing encounters we want to know if we're going to slaughter our party, or just tax their resources a little. For an encounter they are intended to overcome, we don't want a TPK to be the inevitable outcome. If we are willing to use some of the assumptions in the sizing I propose, then we've given ourselves some useful tools.

1) The proposed XP budgets divided over XP to level-up yields about 130 attritional encounters to go from 1-20, or about half that many - 65 - lethal encounters.

2) We can choose a simple tuning value: we can say that at the start of their career each PC should have an N% probability of surviving to level 20

3) We need to make an assumption about revivals: we might say that any time a PC dies, they have a flat 50% chance of being revived (we could improve this by making such an assumption on a per level basis, but let's go with this for now)

Based on the above, if I decide that I want my PCs to have a 50/50 chance of seeing level 20, then very roughly taking our revival assumption into account*, 65 lethal encounters must deliver permanent death. Therefore a lethal encounter could be defined as one with a 1/65 chance of death

*for each character* or about a 6% chance of one or more characters dying in a party of four i.e. about 1/17. (Obviously it is more complicated than this, for example where one character dies it becomes more likely that others follow.)

Thus we could assert that if we run our model 17 times for averagely skilled players (unknown) controlling averagely powerful agents (level = CR?), we should see 1 death. My 1/5 is probably wayyy too high!

*I'm just getting at the principle here. Once we agree a principle, we can fix the maths.