Gambling addictions

Abstract:
If you've ever asked yourself how many tickets you need to have a 65% chance of getting a character to E2S1 (it's 368), or how much it would cost to buy those 368 pulls, or what your chances are of drawing 7 copies of a featured 5* in 7 pulls (the sim says it's less than 5% which is technically correct) then you've come to the right place because I've just spent the last two weeks of my life writing code and running simulations to definitively answer all of those questions and more.
For the impatient among you, here are the two most important tables:
5* Character Banner
5* Light Cone Banner
To use these tables, simply locate the column for the type and level of 5* you're interested in pulling for, then navigate to the row for the appropriate percent chance. For example, looking at the character banner we see there is an approximate 5% chance of getting an E0 featured 5* within 16 pulls. Alternatively, you can look up the number of pulls you have saved and work backwards. To illustrate, if I have 100 pulls then I have around an 85% chance of getting the featured light cone to S1, but only a roughly 25% chance of reaching S2. If you'd like the average case, look at the 50% row (technically this is the median, not the mean/average, but generally with this dataset most measures of central tendency tend to be similar enough that the 50th percentile is a close enough approximation of the average).
A couple of very important caveats regarding these tables. First, some of you may notice that these tables look very similar to those shared in this post by u/Dologue over on the Genshin subreddit. As I'll discuss later, my methodology in generating the above tables differs from that post, but I found their method of data presentation informative enough that I decided to borrow it for this post. Huge shout out to that previous work, without it you would only be getting histograms from me.
I assume a flat 56.4% chance of winning the 50/50 as per this post and data from Star Rail Station. Importantly, my model does not attempt to model WHY this may be the case. I'll talk about this at length below, but for the moment suffice it to say that if you disagree with this assumption, you'll need to either download the simulator from my github and update the rates yourself, or mentally revise the numbers in all 5* tables slightly upward.
These tables assume initial 4* and 5* pities of 0 and that neither the next 4* or the next 5* drawn are guaranteed to be featured. By default stardust is not considered. The simulator is capable of calculating approximate stardust gain, but you would need to download it yourself and enter specific data on the number of characters you own to use that function.
Unfortunately, you can't just add the entries in the above tables to determine your the pulls needed to E6S5 a character for example. Doing that would technically (kinda sorta) give you the number of pulls you would need for a X% chance of getting E6 and a Y% chance of getting S5. Instead, you'll need to go here and scroll to the right until you find the appropriate table. There are simply too many of them for me to post here.
Finally, my modeling of the pity system is based on this discussion on HoYoLAB as well as SRS data. Again if you're interested, I'll discuss this later.
Results:
Okay, here's the part where I pick up the rest of the data like a baseball bat and I hit you over the head with it. Buckle up, there's a lot to get through.
As I mentioned above, here is a spreadsheet with every single relevant table. You can find all the percentile data there, although it may be slightly more difficult to navigate and potentially less entertaining than reading it here on Reddit.
First up, if you're a complete, down-horrendous simp for March 7th like I am and want to know how many pulls it would take to E6 a specified featured 4* character, here's your table:
4* Character Banner (Featured)
If you instead have the Yukong brand of mommy issues, and would like to know how much to save to E6 a specific off-banner 4* character, my sincerest condolences:
4* Banner Character (Non-Featured)
Here is the table for featured 4* Light Cones:
4* Light Cone Banner (Featured)
I genuinely cannot fathom why you would want to save pulls for the sole purpose of pulling for a specific superimposition level of an off-banner 4* light cone. If you are brainrotted enough to actually want to do that, I have made a table for you, but you'll need to go to the spreadsheet and scroll all the way to the right to find it.
I've helpfully generated a cost table to assist you in calculating how much you'll have left over to buy ramen next month. If this isn't sufficient, you can download a CSV that goes all the way out to 2000 pulls in increments of 10 from this link. If that still isn't specific enough for you (seriously?) you'll need to download my simulator and enter the exact number of pulls you want a cost for. (Ignore that buying 50 pulls with the top up bonus is slightly more expensive than without, the function is a bit quirky in how it chooses which packages to buy).
https://preview.redd.it/qvfmalembt3d1.png?width=817&format=png&auto=webp&s=1557cd007bc2ce322e21883f5844bb7dc57f276b
Finally, here are some histograms to help visualize the above tables. The X axis represents the number of pulls to reach the target number of copies of the 5* character of light cone, while the Y axis represents the number of simulations (out of 100,000) that reached the target at that number of pulls.
Figure 1. Number of Simulation Terminated at X Pulls
Methodology:
If you are not a nerd, are not interested in using the simulator tool yourself, and/or are only interested in chancing or costing yourself, all the information you need is contained in the above sections. If you want to learn more about how all that data was generated or why I made certain decisions, read on.
I created a Python program to repeatedly simulate gacha pulls until a break condition was met. That break condition was typically when some target number of featured 5* entities was drawn, but for the 4* tables above I modified the program to stop pulling after the target number of a specific featured or non-featured 4* was drawn. Using a simple loop structure, I repeatedly conducted these simulations. Each simulation reported how many pulls it had taken to reach the specified break condition (along with other data) and I could then compile this information into a list which was used to generate most of the data above.
Technically speaking, the percent chances provided in the tables above are actually percentiles of this list of total pull counts generated by the program. For example, the 50% row represents the cut-point where 50% of all simulations reached the break condition at or before that number of pulls. The 100% column represents the most pulls any individual simulation out of 100,000 took to reach the break condition. This is not technically the maximum possible pulls, but the chances of exceeding it are astronomically low.
100,000 was chosen because further increasing the number of simulation has a significant negative impact on program runtime without appreciably altering the results or improving the fit of the generated data.
Based on Star Rail Station warp data and this statistical model for Genshin, 5* pity starts on pull 66 for light cone banners and pull 74 for character banners. 4* pity starts at 8 and 9 respectively. For every pull past and including these starting points, pity increases by 10* the base 4* or 5* rate for that banner. For example, the base 5* rate for a character banner is 0.6%. On pull 74, that rate would jump to 6.6%, then 12.6% on pull 75 and so on until a 5* is guaranteed.
Discussion:
This Genshin post used a probability density function and constructed a mathematical model to fit the data available from Genshin pulls. I opted to construct a simulator because it allows for more complex control and theoretically more accurate modeling than a function provided the parameters are set up correctly. Essentially, the idea is to replicate the real world process that Hoyo uses to generate gacha results/pull data in the first place. The caveat here is that all of the logic, numbers, and flags used by Hoyo would need to be accurately recreated in the simulator.
The good news is, I've designed it to be extremely modular. Most parameters such as pull rates can be easily adjusted in the file. In theory, through repeated adjustments of rates and other parameters based on data, this program should be able to be tuned to ouput data that very closely matches real world pull results.
I did generate a theoretical probability distribution here, but did not conduct extensive testing to fit my simulated data to it due to time constraints and my lack of access to sufficient real world data.
As I mentioned, the base rate for winning the 50/50 and pulling a 5* character is set to 56.4%. After a limited review of the BiliBili post by OneBST I determined their methodology and attempts to control for response bias to be sufficient for me to use their conclusion in the present study. This 56.4% number is actually slightly lower than the 50/50 win rate reported across multiple banners on Star Rail Station. Notably the above tables were generated under the assumption that the 75/25 win rate for light cone banners is in fact 75%. SRS data would suggest that the actual win rate should be slightly higher, but I do not have access to their database and have not conducted enough analysis to reach a conclusion on this point so I have modeled it as 75% for this simulation to be conservative.
u/Graficat theorizes in this post that win rates are inflated because it is possible to "lose" a 50/50 and have a 1/8 chance of still gaining the featured 5* from the pool of 5* entities available on the banner. If this is correct, it should push win rates to slightly above 56% for the character banner and slightly above 78% for the light cone banner, both of which are in line with the SRS data. Further, they posit that losing the 50/50 while still gaining the featured 5* can result in the next 5* being guaranteed as well (assuming the system based the guarantee off of whether the last 50/50 was lost as opposed to whether the last 5* character obtained was featured).
The present simulation DOES NOT account for this, again to be conservative. For 99% of use cases, especially as a guide for saving pulls, this should be fine. Currently the simulation can only award a featured 5* if the 50/50 (at 56.4% odds) is won, and winning the 50/50 sets the guarantee to False. Losing the 50/50 (at 44.6% odds) can never award a featured 5* and always sets the guarantee to True. Moving forward I plan to model the above suggestions to see how the data and 50/50 win rates are affected by handling the guarantee and win rates in various ways. This is part of the iterative modeling process I describe above.
Conclusion:
It is my hope that you found this post educational, entertaining, and/or useful. If you would like to double check my work or play around and iterate on the simulator I've produced, you can download it here. Please note that you will need Python installed on your system to run it; I am trained in using Python for data analysis and have no idea how to package nice little applications or web interfaces. Let me know if you run into any issues.
I promised to throw together a simulation for a since deleted user over on FireflyMains about 9 days ago. From start to "finish" this project has eaten the better part of those 9 days. If you'd like to support my work and are interested in seeing more of this stuff from me in the future, I've made a Ko-Fi where you can help fund my gambling addiction field research.
This manuscript submitted to the Belobog Ministry of Education for approval and publication.