Modeling single pulls
Lijia Wang
2020-12-16
Last updated: 2020-12-17
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Knit directory: gacha/
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Introduction
In this section, we will introduce and model a gacha scenario where all characters in the same rarity level have the same probability of getting pulled, and pulls are performed one at a time (therefore no bonuses are applied). We would like to find the expected number of trials until we obtain any SSR character.
Method
Pulls can be modeled as a Bernoulli Process
In this scenario, the probability of getting each tier is constant, and the probability to pull each character within the tier is the same. Let the probability of pulling an SSR be \(p\), and suppose there are \(n\) characters in the SSR tier, then the probability to pull each individual SSR character is \(\frac{p}{n}\). We also assume that pulls are performed one at a time and under the assumption that no previous pull affects the result of the next pull.
This process can be modeled as a simple Bernoulli process with an unfair coin, where probability of obtaining an SSR character (“success”) is \(p\) and not obtaining an SSR (“failure”) is \((1-p)\). This means we would probably expect on average \(1/p\) pulls to obtain the first SSR.
Poisson Distribution
We notice that the probability of success \(p\) is typically very small in each gacha game, ranging from 0.600% (Genshin Impact) to 2.000% (Arknights) in the examples we use. Therefore, we are safe to approximate the probability of SSR pulls with a Poisson distribution.
\[ f(k; \lambda) = Pr(X=k) = \frac{\lambda^k}{k!}e^{-\lambda} \]
Since \(p\) small, we can approximate \(\lambda = np\). We can subsequently rephrase our question into “in how many trials can we maximize the probability of obtaining exactly 1 SSR card”?
Application
We plotted the probabilities of obtaining 1, 2, and 3 SSR cards in 1 to 500 pulls. Here we used the Fate/Grand Order Saint Quartz Summon Distribution Rate on an non-event banner. We define “success” as pulling a 5-star servant (character), and the probability is 1.000% (according to in-game statistics provided).
[1] "Number of pulls that maximizes probability of 1 SSR is 100"[1] "Number of pulls that maximizes probability of 2 SSRs is 200"[1] "Number of pulls that maximizes probability of 3 SSRs is 300"
| Version | Author | Date |
|---|---|---|
| 998f5f3 | Lijia Wang | 2020-12-16 |
Remarks:
We can observe that performing single-pulls alone is not very statistically interesting. Although it is fun to see that if you pulled 100 times and still did not get one single SSR card… you’re truly quite unlucky.
R version 3.6.2 (2019-12-12)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
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attached base packages:
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