Demiatma Probability

How many dudes do I have to kill to get some of these stupid jolly ranchers?

TL;DR: if you want at least a 90% chance of getting at least one demiatma drop, you’ll need to be prepared to do eleven Critical Engagements or forty-five FATEs.

The Problem

I am a big fan of the critically acclaimed MMORPG Final Fantasy XIV (with an expanded free trial etc. etc.). I am not however a big fan of grindy gameplay, and FFXIV serves up a heaping helping of that with its new Occult Crescent content, specifically the demiatma grind, where you have to kill dudes in order to get three each of these stupid things:

The six flavours of demiatma on offer in the Occult Crescent and Dawntrail FATEs

It’s not that I don’t like things that are challenging or take a long time to do. I just don’t like things that are arbitrarily difficult and needlessly repetitive. If I decide to devote time to some effort, I like to know roughly what I can expect to have accomplished at the end of doing so.

The RNG involved in these kinds of grinds makes that difficult, especially when the drop rates are low. This past week, we learned in the Live Letter that the drop rate on these particular items is 20% for Critical Enagements and 5% for FATEs. Not great, but at least with hard numbers we can start to quantify what that actually means in terms of effort versus reward.

Basically, when I sit down to play, I want to know that if I put in some fixed amount of effort, I have some acceptably high chance of seeing progress in the form of at least one demiatma drop. How do I know what that fixed amount of effort should be? We can figure that out with some math.

The Maths

At a high level, we could compute the probability of some number of successes (drops) n across some number of attempts (completed FATEs or Critical Engagements) k with some probability (0.05 for FATEs, 0.2 for CEs) p with a binomial tail probability:

\(f(k, n, p)=\begin{pmatrix} n \\ k\end{pmatrix} p^{k}q^{n-k}\)

Where “n choose k” is our binomial coefficient, defined by:

\(\begin{pmatrix} n \\ k\end{pmatrix} = \frac{n!}{k!(n-k)!}\)

Given the p values we’re working with in this case are quite low, however, I think it’s easier to ask the question “what’s the probability of seeing at least one drop across some number of attempts” and extrapolate that out to see how many attempts we’d need to take at CEs or FATEs to make the smallest possible unit of progress.

We can do that with the complement rule of probability theory, which results in the considerably simpler-to-compute:

\(P(\text{at least one drop}) = 1 - (1 - p)^n\)

Where p is our probability (0.05 or 0.2) and n is the number of attempts we’re making. As an example, for one CE, we already know the answer—it’s 20%. But if we tried five times?

\(P(\text{at least one drop}) = 1 - (1 - 0.2)^5 = 1 - 0.32768 \approx 0.672\)

So we’d have about a 67% chance of getting at least one demiatma if we did five CEs. Okay, but not great. Rather than calculate them all out one-by-one until the numbers are to my liking, let’s say I want to know how many attempts it would take me to have a certain target success probability. We can figure that out with:

\(n \geq \frac{\log(1 - P_{\text{target}})}{\log(1 - p)}\)

So if an individual attempt has a 20% chance of success and we want to know how many attempts we’d have to do to have a 90% chance of at least one success, we can now compute that as follows:

\(\begin{aligned} n &\geq \frac{\log(1 - 0.9)}{\log(1 - 0.2)} \\ &\geq \frac{\log(0.1)}{\log(0.8)} \\ &\approx \frac{-1}{-0.09691} \\ &\approx 10.3 \end{aligned} \)

Therefore, to have a 90% chance of at least one drop, you need to do at least eleven CEs. (If you’re curious, you’d have to do at least forty-five FATEs to achieve 90% likelihood that route).

The Result

I haven’t timed it yet, but I’d ballpark that a CE spawns maybe every 10 minutes or so. If that’s close to true, it means when I sit down to play, I need to be prepared to dedicate about two hours to running CEs if I want a reasonable expectation of progress. If I have less time than that, I probably won’t bother with it, given the numbers above.

So that’s my breakdown—if you find yourself also on the demiatma grind, good luck, may RNGsus smile upon you, and stay safe out there in the wilderness.