Genshin Probability Deep Dive
Version: v3.10.0
Wishes
A pull or roll is referred to as a "wish" in Genshin Impact lingo. A wish is a random event resulting in an item of varying rarity and quality.
Banners
In Genshin Impact, players can choose to wish on different banners. These banners have unique mechanics and possible items. Many guides exist on the details. Here we focus on the probabilistic mechanics of these banners.
Pity
Genshin Impact, like many gacha games, has a pity mechanic that guarantees a hard limit on the number of wishes to attain a certain level of item (not necessarily a specific item). The advertised hard pity limits are 90 wishes for a five star on standard or character banner, 80 for weapon banner and 10 for four stars on any banner. The pity mechanic is where most of the community speculation and investigative efforts are focused. Specifically, the details (or even the existence) of the "soft pity" are not published by miHoYo and must be derived from wish data.
Theories on soft pity
[1] Excellent folks in the community have already performed analysis on the soft pity:
https://www.reddit.com/r/Genshin_Impact/comments/jo9d9d/the_5_rate_is_not_uniform_06_there_is_a_soft_pity/[2] With the primary source of data coming from "whale watching" or videos of folks wishing a ton.
https://www.reddit.com/r/Genshin_Impact/comments/jod9o4/whale_watching_logs_2_the_blue_whale/From these sources, we believe 1) there is a soft pity 2) the soft pity mechanic is a linear increase in probability starting at some threshold and hitting 100% at the hard pity limit.
How to sanity-check soft pity theories
How can we check our theory? miHoYo publishes base and consolidated rates for every banner and level of items. For example, the consolidated rate for a five star on the standard banner is 1.6%. The base rate is merely 0.6%. Let us check one possible theory of soft pity - that it doesn't exist.
Suppose every wish from 1-89 has probability 0.006 (0.6% base rate) until the 90th which has 100%. Does this give us a 1.6% consolidated rate?
If the consolidated rate is 1.6% this implies the average chance of getting a five star on any wish is 0.016. The expected number of wishes it would take to get a five star is then:
\[\mathbb{E}[\text{wishes for five star}]=\mathbb{E}[Geom(0.016)]=\frac{1}{0.016}=62.5\]The general distribution for trials until you get something is known as Geometric with a well-known expected value: https://en.wikipedia.org/wiki/Geometric_distribution#Expected_Value_Examples
Now let us check whether only hard pity gives us an expected value of 62.5.
\[\displaylines{\text{Let } X = Geom(0.006) \newline \mathbb{E}[\text{wishes for five star with no soft pity}]= \mathbb{E}[X | X < 90] (1 - (1 - 0.006)^{89}) + 90(1 - 0.006)^{89} = 69.69969 \gt 62.5}\]In English: Expected wishes to get a five star with 0.006 base rate and hard pity = (expected given it took less than 90 wishes) x (probability it took less than 90) + 90 x (probability it took 90 aka hard pity)
Without soft pity, our expected wishes it takes to get a five star is too high. This means our consolidated rate is too low. There must be something beyond just hard pity to get the 0.016 consolidated rate (and thus 62.5 expected wishes per five star). This could be a flat probability increase starting at some pull number, a linearly increasing probability or even some non-linear ramping function.
Five star soft pity: linear ramping
Our favorite theory is linearly increasing probability starting at the 74th wish and reaching 100% at the 90th. This results in a ~1.60% consolidated rate per this simulation: [3] https://bbs.nga.cn/read.php?tid=23711258&rand=868. We can also numerically integrate to get an exact answer of 1.604% (or 62.338 expected value). This does not exactly match the 1.600% advertised by Genshin Impact. However, we do not believe it is worth further tweaking the probabilities to get exactly 1.600%. The simplicity of linear ramping is compelling and the graph matches the histogram from [2]. The same pity system is used in games like Arknights (mentioned in [1]).
Four star soft pity
Much less community research exists regarding four star pity. Our original source [1] mentions a soft pity starting at the 8th pull. We decided to do our own investigation. [4]
https://www.reddit.com/r/Genshin_Impact/comments/kp8p96/four_star_hard_pity_on_weapon_banner_is_basically/From our data, four star pity on standard banner starts at the 9th wish (8th for weapon). Lets use the consolidated rate analysis from above to take a guess at the soft pity mechanics.
Calculation with no four star soft pity (standard banner)
\[\mathbb{E}[\text{wishes for five star}]=\mathbb{E}[Geom(0.13)]=\frac{1}{0.13}=7.69\]\[\displaylines{\text{Let } X = Geom(0.051) \newline \mathbb{E}[\text{wishes for four star with no soft pity}]= \mathbb{E}[X | X < 10] (1 - (1 - 0.051)^{9}) + 10(1 - 0.051)^{9} = 7.990 \gt 7.69}\]Again, without something above the hard pity, our rate is too low (expected wishes to get a four star is too high).
Four star soft pity starting at 9th wish
Let us consider a soft pity where 9th wish gets 0.5 probability and 10th wish gets 1.0, the rest have the base probability of 0.051
\[\displaylines{\text{Let } X = Geom(0.051) \newline \mathbb{E}[\text{wishes for four star with 0.5 at 9 and 1.0 at 10}]= \mathbb{E}[X | X < 9] (1 - (1 - 0.051)^{8}) + 0.5 \cdot 9 (1 - 0.051)^{8} + 0.5 \cdot 10 (1 - 0.051)^{8} = 7.6955 \approx 7.69}\]The expected number of wishes it takes actually lines up with the 13% consolidated rate. Supporting this further, data from [4] strongly suggests soft pity does not start until the 9th wish. Given all this, our calculator uses 0.051 until 0.5 at the 9th wish and 1.0 at the 10th. An exact table can be found on the Inferred Probabilities page.
Weapon banner four star soft pity
Similar arguments can be made for the weapon banner, which has a 14.5% consolidated rate. Our data [4] show that significant pity starts on the 8th wish and hard pity is effectively the 9th wish. This differs from the stated 10 wish hard pity in the game (well - to be more accurate they say "less than 10 wishes" so they could be technically correct).
The 50/50 and Capturing Radiance
Version 5.0 added a mechanic that can quietly turn a lost 50/50 into the promotional character, and miHoYo published exactly two numbers about it and nothing else. A five star won while you're not on a guarantee is the promotional character 55.000% of the time, and once the promotional character has been your second five star on three consecutive occasions the conversion is guaranteed. How it actually works isn't published.
Two constraints and one unknown is enough to work with though. Say the conversion fires with the same probability c on every lost 50/50 and is forced once you've lost three in a row, and let q = 0.5(1 - c) be the chance a 50/50 ends up as a loss that didn't get converted.
\[\frac{1}{1 + q + q^2 + q^3} = 0.55 \implies c = 0.04212\]So a 4.212% conversion chance, rescuing about 8.4% of the losses you'd otherwise take, and it's the only constant trigger rate that's consistent with both published numbers. That's what our calculator is using. It isn't a fact about the game though, it's what follows if the trigger doesn't depend on how long your streak is, and nobody has actually established that.
Most calculators skip all this and use a flat 55/45, which gets the mean right and the tail wrong (a flat 55/45 will happily lose you four in a row 4.1% of the time and the official rule says that can't happen). GGanalysis goes the other way and models only the streak guarantee, which gets 53.33% and overstates the wishes you need a little, and their own source comments say as much. We'd rather match both published numbers than one of them.
Oh and there's a third published figure we can't do anything with. miHoYo also gives a 0.018% base probability of triggering Capturing Radiance, and no reading of it we tried gets us to 55.000% (the tempting one gives c = 6% and a 55.72% share). We left it alone rather than pretend, same as we did with 1.604% against a published 1.600% further up this page.