Roll Probability

Introduction

For many Genshin Impact players, they’ve experienced such a poor chain of luck in terms of their Wish banner rolls at some point in their play that they’ve started asking the question of whether or not the probability in their pulls was “rigged”. In other words, that what Mihoyo has advertised as a drop rate for 4- and 5-star items isn’t what they’re experiencing firsthand in their play.

If I’m being honest, there was a point in my gameplay where I asked the same question, and was looking for resources to try and provide an answer on the burning question of: “Is the roll system in Genshin Impact unfairly rigged for certain players?”

In the advent of not being able to find many other resources on the Internet that holistically walk through the entirety of how to statistically, mathematically calculate roll probability for not only Genshin Impact but for other gacha games as well, this guide was created as a one-stop-shop for all the information you need to perform an investigation of your own account and rolls to see what your “luck” actually is.

In this guide, we’ll walk through some of the common misconceptions about roll probability, go over some of the information that we can only make educated guesses on, and then get into some down-and-dirty math to showcase how you can find out what your roll probability actually is, and whether or not it adds up to what Mihoyo has advertised it to be.

It should be stated that this is the most exhaustive Math & Mechanics guide on the Sons of Dvalin website, as well as being the biggest by far. Be prepared for a long read, and the possibility of needing to re-read certain sections. Take things at a measured pace.

Misconception #1 – The “Block-Style Pull”

Speaking for myself, I remember going to county and state fairs when I was young and seeing some of the “Test of Wits” or “Test of Skill” booth-style challenges that tended to be set up near the amusement park area of the fairgrounds. In some of these challenge games, the organizer would place a small object (such as a marble) under one of five upended paper cups, then quickly move the cups around so as to confuse the player on which cup the object was under. The player then had to guess which of the cups the object was under, all the while knowing for certain that the object was indeed under one of them.

Many people mentally take the same concept and apply it directly over to gacha games. For example in Genshin Impact, saying that a 10-pull on a banner always has at least one four star hidden within one of the rolls in that “block” of 10 pulls, and it’s just a matter of finding it.

While this is a very understandable way of explaining how a gacha hard pity counter works for a set of pulls, it’s unfortunately not factually accurate. Here’s why:

  1. First off, Mihoyo expressly states that this is not the way that rolls work in Genshin Impact. If you open up a limited character banner, for example, and click on the Details button for it, you will be told in the Rules section that you have a 5.100% base probability chance on any given pull that that pull will yield a 4-star drop. This implies that raw rolls are based off of RNG in line with this percentage, and not with a “block-style guarantee” pull system.
  2. The block-style pull mentality doesn’t account for the idea of soft pity, and the playerbase for Genshin Impact can say with 100% certainty that there is such a thing as soft pity when rolling for a 5-star drop. If probability on drops was a static thing as with the block-style pull idea, there would be no need for a soft pity mechanic with 5-star pulls.
  3. From a programming standpoint, coding a block-style pull mentality in line with the concept of soft pity would be a nightmare to work out the logic on, and would be very resource-intensive in terms of its computation. By contrast, a base probability chance system with an incorporated soft pity mechanic is much simpler to code and maintain.

Misconception #2 – We Know Exactly Where “Soft Pity” Starts

About two months into Genshin Impact being a live game to the public playerbase, several sources performed calculations on large chunks of player statistics in pulls, and determined that based on the data, it was indisputable that there was some kind of mechanic coded into Genshin that started significantly increasing relative probability on pulls for a 5-star drop once you hit a certain roll threshold without yet getting a 5-star. This data was corroborated by several other sources over time, and other independent studies.

Mihoyo also stated this mechanic existed after announcing that the consolidated 5-star probability rate on pulls between an entire count of 90 pulls is 1.600% – meaning that base pull probability does increase at some point in the course of your rolls in that count of 90. How much does it increase by, and where at? Those two variables are still mysteries since data mining can only get us close to the actual numbers, and Mihoyo has never formally said what they are.

What’s also still somewhat up to general debate at this time, is where soft pity actually begins in terms of roll count, as the studies all land in the same general ballpark, but there’s a degree of natural variance in their results within that ballpark, likely due to standard deviation.

Some sources confidently say that soft pity begins at 70 rolls, and the data they present is well-organized and very believable. Others argue that it begins at 73 rolls, with similar datasets. Still others present a case that it begins at 75 rolls, and have their own small mountain of information to back their claim up.

Ultimately, these numbers are all so close, relatively speaking, to each other that they’re likely falling within the realm of what standard deviation can accommodate. We’ll touch heavily on standard deviation later in this guide, so don’t worry – we’ll go over a depth explanation of what it refers to.

Presenting The Logic, Math, and Tools

The first thing we’re going to need to do is determine what your role probability for several different things in Genshin is supposed to be, based on what Mihoyo has formally published on it. What we know for a fact out of the gate is that:

  • Limited Character Banner
    • Probability of pulling a 4-star on an individual roll outside of 4-star hard pity is 5.100%.
    • Probability of pulling a 5-star on an individual roll outside of 5-star soft pity is 0.600%.
    • Hard pity on pulling a 5-star is set at 90 pulls.
  • Limited Weapon Banner
    • Probability of pulling a 4-star on an individual roll outside of 4-star hard pity is 6.000%.
    • Probability of pulling a 5-star on an individual roll outside of 5-star soft pity is 0.700%.
    • Hard pity on pulling a 5-star is set at 80 pulls.
  • Standard Character Banner
    • Probability of pulling a 4-star on an individual roll outside of 4-star hard pity is 5.100%.
    • Probability of pulling a 5-star on an individual roll outside of 5-star soft pity is 0.600%.
    • Hard pity on pulling a 5-star is set at 90 pulls.

Since it’s up to debate as to where the soft pity mark for 5-star pulls actually sits at, for the sake of our discussion, we’ll steer on the safest side of the spectrum that we can, and assume that soft pity begins at 70 pulls on a character banner, and 60 pulls on a weapon banner. As such, we’d only calculate for rolling up to the 69 and 59 pull marks on those banners, respectively.

Similarly, in an effort to stick to only solving for probability and not solving for something that’s certain, we’ll only run math for 4-star characters on the 9 pulls leading up to 4-star hard pity, and not for the 10th pull that is the guaranteed pity.

Before we proceed further, I’d like to introduce two tools to you that will dramatically help in speeding up our calculations and automating a lot of the complex math we’d otherwise have to manually do.

Gacha Chance Calculator – https://www.dskjal.com/statistics/chance-calculator.html
Sample Size Calculator – https://www.calculator.net/sample-size-calculator.html

These tools are the best ones I’ve found in terms of handling the things they do. I’ve also run their math against mine, and compared it to lesser-made tools produced by others, and the math consistently lines up across all fronts. The data you’ll get from these tools can be trusted to the best of my ability to test.

So, what are we going to do with them? Well, we ultimately need two things. The first thing we need is to determine what the additive probability is for getting 4-star and 5-star drops outside of pity. In layman’s terms, this is the natural odds in a set of 9 rolls that you have of pulling a 4-star, and the natural odds in a set of 59 or 69 rolls of pulling a 5-star. We aren’t going to factor hard and soft pity into the probability equation since hard pity isn’t a matter of probability in the first place, and soft pity involves a variable percentage scaling that we don’t know absolute numbers for – making it unusable in terms of our calculation.

The second thing we need is to determine how large of a sample size we have to assemble in order to gain reasonable certainly that our calculation is correct, and also to reduce standard deviation. Put simply, how many sets of 9 rolls for a 4-star, or 59/69 rolls for a 5-star do you need to actually determine if what you yourself are actually seeing on your own rolls actually lines up to the chance probability that Mihoyo has advertised for those rolls.

Calculating Additive Roll Probability

To solve for this, we’ll use the Gacha Chance Calculator tool mentioned in the above section. This is a marvelously well-made tool that can actually applied to almost every chance gacha game in existence – not just Genshin Impact.

To use this tool, scroll down to the Chance Calculator section. You’ll see it present information similar to the following:

For an example, let’s plug in the information we know about a set of 9 rolls towards a 4-star character from a limited character banner pull set. Doing so, we’d get:

Very cool! This means out of a set of 9 rolls towards getting a 4-star drop on a character banner, we’d have an overall chance of 37.750% in getting a 4-star outside of the hard pity roll that would have been our 10th.

If we solve for the remainder of the probability sets we’re interested in, we get:

  • Limited Character Banner
    • Probability of pulling a 4-star in a set of 9 rolls is 37.750%.
    • Probability of pulling a 5-star in a set of 69 rolls outside of 5-star soft pity is 33.982%.
  • Limited Weapon Banner
    • Probability of pulling a 4-star in a set of 9 rolls is 42.701%.
    • Probability of pulling a 5-star in a set of 59 rolls outside of 5-star soft pity is 33.930%.
  • Standard Character Banner
    • Probability of pulling a 4-star in a set of 9 rolls is 37.750%.
    • Probability of pulling a 5-star in a set of 69 rolls outside of 5-star soft pity is 33.982%.

Alright! So, in general, we’re hovering either at or slightly above a 1 our of 3 chance that on any given character banner, we’ll be able to pull either a 4-star or a 5-star outside of the respective hard or soft pity for that drop. The weapon banner sits slightly higher for 4-stars, at just over a 2 out of 5 chance.

What we’re now going to need to supplement this information with is an understanding of how many sets of pulls we’d have to have in order to check if our own “luck” in pulls is actually on track with these numbers.

Calculating Sample Size & Standard Deviation

There’s an art to figuring out how many sets of something you need to be able to statistically identify a trend, but as with most things in the magical realm of statistics, there’s more to the story than just that. Sample size as a variable doesn’t live alone – it has to exist together with how much percentage certainly you have that a moderate percentage of your sample set is going to align to what your statistical average should be, and how much deviation items within your set can have to still be considered “close enough” to your average.

It’s a difficult concept to explain in words, and is easier to understand once you see it in action – so let’s do that.

Using the Sample Size Calculator that we mentioned previously, we’re presented with the following pieces of information. They can be used a couple of different ways, which we’ll go over in turn:

We need to find out how big our sample size of pull data needs to be in order to gain reasonable certainty that our own personal pulls line up to what Mihoyo states that they should be, or if our pull luck is below or above average. To use this tool to determine this, the top set of values we’ll test out are:

  • Confidence Level
    • This is how much of a given sample set is going to fall within the bounds of being “close enough” to the average or absolute pull probability, after we’ve accounted for standard deviation.
  • Margin of Error
    • This is the same as standard deviation. It’s a relative percentage of how far off a given example set can be from the mark to still be considered “close enough” to be within the bounds of what we’d expect our outcome to be. Important to note here that this is a relative percentage, not an absolute percentage. For example, if you added 5% relative percentage to 50%, you’d get 52.5% total because 5% of 50% is 2.5%, not 5%.
  • Population Proportion
    • What this means in regard to our purposes is the amount of possible answers you can have in a given outcome. Always leave this at 50% for everything we’re doing here. What 50% in this field means is that we only have two possible outcomes – yes, or no. This is what we want because we’re either getting a 4-star character in an example roll, or we’re not; same for 5-stars within the scope of our data.

So, in order for me to find out how large of a sample set of 9-rolls towards a 4-star drop on a character banner I’d need, in order to be 95% sure that I was within 5% margin of the luck Mihoyo has stated I’m supposed to have in pulling for 4-stars is, I’d get:

Remember how we saw in our previous section that the odds of pulling a 4-star drop in a set of 9 rolls on a character banner was 37.750%? Well, let’s factor +/- 5% relative percentage onto that number using the below equations to find out what our standard deviation (error margin) is on that number:

37.750 x 0.05 = 1.8875

37.750 + 1.8875 = 39.6375% / 37.750 – 1.8875 = 35.8625%

So, if we have a sample size of 385 samples of us rolling for a 4-star drop in 9-roll sets on a character banner, 95% of Genshin Impact players should have a probability between 35.8625% and 39.6375% of pulling a 4-star drop outside of hard pity on those roll sets. If your probability sits below this, your luck is abnormally bad. If yours is above this, your luck is abnormally good. In either abnormal outcome, you’re within the 5% of players that “break the norm” in terms of your pull rates.

While this is cool to see, how many players have actually done 385 sets of 9 rolls on a character banner? Quite a few players haven’t done that many. If you’re one of those players, you may be asking the question of how to make this data actually usable for you.

To do this, you’ll need to open up the banner in question that you’re wanting to calculate for, and start through the roll History from the very beginning of your rolls on that banner. Using 4-star characters as our example, start adding up the number of 9-roll sets you had where you pulled a 4-star outside of hard pity, vs. the times you didn’t. Just remember in doing this that:

  • You have to reset your count towards 9 in a set once you pulled a 4-star.
  • Rolls within the soft pity range towards a 5-star cannot be counted, as Mihoyo has stated that consolidated probability exists for 4-star pulls as well.
  • If you pulled a 5-star, reset your count towards 9 in your set, but do not count the 5-star in your 4-star hit/miss total.

Let’s suppose that I went through my player profile for limited character banners, added everything up, and found that I had 75 counts of 9-roll sets towards pulling a 4-star, and I pulled a 4-star outside of hard pity on 24 of them. Or, in other words, I had a 32% pull rate for 4-stars outside of hard pity in 9-roll sets.

Using the second half of our Sample Size Calculator tool, I can plug this information in using as much of our first example’s data as I can, and get the following output:

A much higher standard deviation percentage than we first witnessed! This should be expected, though, since we’re dealing with a much smaller sample size and our error margin is subsequently going to be larger to compensate. Plugging this into some quick math to find out what our actual percentages are, we get:

37.750 x 0.1132 = 4.2733

37.750 + 4.2733 = 42.0233% / 37.750 – 4.2733 = 33.4767%

Interesting! So, in this example, my account would actually fall within the 5% of Genshin Players who have had abnormal “luck” in their rolls, as my 32% pull rate for 4-stars outside of soft pity is actually below the 33.4767% minimum percentage rate that standard deviation can actually accommodate. Otherwise, 95% of players who’ve had at least 75 sets of 9-rolls towards a 4-star on a character banner should see between a 33.4767% and 42.0233% pull percentage on 4-star drops outside of hard pity.

It’d suck for me, but mathematically, some players do have incredibly bad luck such as this in their rolls. And my luck is apparently below the minimum allowed for standard deviation…

Observations

You can use these tools and this data for several different things.

These principles don’t apply to just Genshin Impact; they can be applied to the lion’s share of gacha games currently available in the consumer marketplace. I’ve tested them a bit on Arknights and found them to be fairly accurate for my former play experience for it, and have seen reports that examples used with Fate: Grand Order have also held true.

They can be useful for individual players looking to see where they actually sit in terms of their roll luck vs. what they should expect to see in terms of the general player landscape. On the author’s part, when I used these for myself in the creation of this guide, I found that a longstanding suspicion I’d had that my account had below-acceptable luck in terms of my pulls was actually just a suspicion – my luck was actually within the realm of statistical norms.

If you were particularly inclined, you could also use this data with a large enough playerbase to try and analyze whether or not Genshin Impact’s reported stats could be proven to be false within reasonable statistical confidence. In other words, you could use them as a means of being able to back up conspiracy theories on whether or not the game truly is rigged on a somewhat large scale.

Regardless of what you use them for, it’s our hope that in presenting the tools to you, showing our math and examples, and illustrating applications of them, we’re giving you a long, but better demonstration of how to determine these pull stats for you or for others. Extremely few guides on the Internet adequately explain how these mechanics work, and far too many of them don’t show the math or tools behind how they arrive at the numbers they do.

Conclusion

It’s our hope that this guide has been helpful for you, and you’ve found it insightful for your playstyle and general approach to the game.

We welcome comments or constructive feedback if additions or corrections can be made to to this content. Feel free to drop us a note with your thoughts.

Cheers,

The Sons of Dvalin community