Gacha rates(stupid question)

this sounds stupid but is been eating away at me

so I’ve been reading in the summon info and read that the rate for a 5 star Character is 1%

lets look at the ishtar banner it it says the same info that theres a 1% chance for a 5 star and 3% for a 4 star

and after scrolling down you see that ishar has a 0.70% drop rate and others have 0.035% rate

so how is the character you get realy determined?

is it
A) that you have a 1% chance to get a “5 star character” and after you get that 1% to get a 5 star you have a 0.70% chance to get the rate up character(in this case ishtar) and a 0.035% to get any other 5 star

or

B) that every roll you get a 0.70% to get the rate up character and 0.035% to get a spook

I’m kind leaning to B but whats with the “1%” thats said on the begining? did they simply round it up?

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the entire roll is finalized once you hit summon, so the distinction is meaningless

also, it’s .7% rate up .3% spook for ssr servants

or .4% rate up for double rate up banners, iirc

idr triple rates

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Rate up is a lie.

I got 2 Arjunas, 1 Napolean, 3 Fragments of 2030, 1 Black Grail all before I got my first Abigail.
The chances of that happening are so astronomically low that it is essentially impossible if pulls are random but possible if pulls are not truly random.

low chance is still not no chance

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oh I see so thats why its 1% thanks

well in my experience I have 7, 5 stars and out of all of them only one was a spook, so Rate up isn’t a lie(for me atleast)

It’s A. Firstly, the game determines the rarity of a card. If it hits 1% (you get 5 star servant), then it decides to pull rate-up servants or non rate-up.

When it comes to gambling, the odd must be specified: DW is not allowed to do the hidden math game here. So they have the table with precise chance for everything you could get in the Summon Info:

  • Ishtar: 0.7%
  • Altria, Mordred, …: 0.015%

If you sum up all the remaining 5* (there are 20 of them), you should get 0.3% (or just take 0.015% x 20). So the total chance to get an SSR (either Ishtar or a spook) is 0.7% + 0.3% = 1%.

Nope, far from impossible. Each roll is an independent event.

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Each roll is an independent event.

I highly doubt that is the case.

The gacha uses pseudo-RNG which is far from independence. However, it is practically better for DW since (if implemented correctly) PRNG can guarantee the advertised rate statistically (in other words, if we collect the summon results from every players, we do get roughly 1% of SSRs), hence

  1. predictable revenue stream and
  2. the players are ensured some luck (though some are much more blessed than others). This in turn ensure that a large number of players do not quit due to salt.
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My understanding is that computer RNG is at best simulated, but for all intents and purposes, we should consider it to be RNG.

DW would be putting themselves at legal risk if they were ever discovered to have been putting their thumb on the scale in any undisclosed manner.

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DW would be putting themselves at legal risk if they were ever discovered to have been putting their thumb on the scale in any undisclosed manner.

They wouldn’t do this of course. My idea was that: don’t automatically assume independence in real life.

RNG by itself works in 2 different contradicting directions

  1. It should ensure unpredictability e.g. seeing past rolls don’t help predicting the next one
  2. It should offer some predictability in the form of rate and distribution.

It is not possible to ensure both. I believe real life lottery (e.g. Powerball) uses atmospheric pressure as a source of randomness which is great for (1) but not for (2): How do you know the chance of getting a particular pressure value is equally likely everyday? Doing this is OK for lottery since there is no guaranteed rate.

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This is all kinda off-topic, but true randomness is possible to generate via radioactive decay. By measuring how long it takes for an unstable atom to decay, you can generate truly unpredictable, random numbers.

As for RNG being both predictable and unpredictable:
Imagine you roll two 6-sided dice, one colored red and the other colored blue. There are 36 possible distinct combinations, 6 for the red die * 6 for the blue die, and each roll is essentially independent of the previous rolls (assuming stuff like ideal dice, perfect unpredictable rolls, etc.)
But if you record your results over a long period of time, you find that the sum of the numbers on the two dice is most likely to be 7. In fact, with an infinite number of trials, we’d expect that 7 would be the sum 1/6 of the time.

What seems to be tripping you up here is the Gambler’s Fallacy: while reversion to the mean is the tendency given a sufficiently large number of rolls, each individual roll is still unpredictable. After flipping a coin and getting 5 heads in a row, I’m not more likely to get a tails instead of a heads on my next flip, even though the probability of flipping 6 heads in a row is tiny: 1/64. That’s because the sequence of 5 heads and then a tail is equally tiny: 1/64.

So ultimately, random number generation is truly random, and you can consistently produce values that are impossible to predict. Given a sufficiently large number of trials, you can ensure relatively uniform distribution, but extremely unlikely but possible events can still happen. I’m not privy to how DW produces gacha rolls, but they most likely generate some big random numbers between 1 and 100,000 or so, and then look up each value in a big table, where each number is tied to a specific value (Mapo Tofu, Dragon’s Meridian, Arjuna). Then, they return the results to you in your daily dose of salt.

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That’s not astronomically low at all. This is a pretty basic probability problem:

4 rate-ups, 0 spooks
.7 * .7 * .7 * .7 = .2401

3 spooks, 1 rate up (.0189*4 = .0756)
.3 * .3 * .3 * .7
.3 * .3 * .7 * .3
.3 * .7 * .3 * .3
.7 * .3 * .3 * .3

2 spooks, 2 rate up (.0441*6 = .2646)
.3 * .3 * .7 * .7
.3 * .7 * .3 * .7
.7 * .3 * .3 * .7
.3 * .7 * .7 * .3
.7 * .3 * .7 * .3
.7 * .7 * .3 * .3

3 rate up, 1 spook (.1029*4 = .4116)
.7 * .7 * .7 * .3
.7 * .7 * .3 * .7
.7 * .3 * .7 * .7
.3 * .7 * .7 * .7

4 spooks, 0 rate up
.3 * .3 * .3 * .3 = .0081

You don’t have to list out every possible outcome like I did, you can just use combinations like 4C2 for 2 spooks, 2 rate-ups, and multiply that with the individual probability.

Anyway, your chances of pulling of pulling 3 spooks and 1 rate up out of 4 SSRs is 7.56%. That’s not impossibly low, though the odds of specifically pulling 3 spooks AND THEN 1 rate up is only 1.89%. Pretty bad luck.

But compare that to my odds of going over 2200 quartz on my alt without pulling a SSR ever – let’s say that’s about 734 rolls. The odds of that are a mere .99^734 … or .06255%. Now THAT is what you would call astronomically low.

that’s not how probability works thanks

They wouldn’t do this of course. My idea was that: don’t automatically assume independence in real life.
RNG by itself works in 2 different contradicting directions

  1. It should ensure unpredictability e.g. seeing past rolls don’t help predicting the next one
  2. It should offer some predictability in the form of rate and distribution.
    It is not possible to ensure both.

I don’t buy this. There is no contradiction between having a random, unpredictable event following a predictable probability distribution. This applies equally well with real life random events such as rolling dice, drawing cards, or radioactive decay as it does with RNGs. You don’t need to apply some sort of Gambler’s Fallacy correction to smooth out the distribution and ensure that predictable rates are met, the law of large numbers will ensure you get your expected distribution without needing to compromise the unpredictability of any given pull.

Certainly if you know the exact algorithm and seed used you can predict the outputs of a RNG, but you aren’t going to have that information, nor are you going to see systemic differences from a truly random distribution such as “if your last pull was an SSR, the chances of you next pull being an SSR as well are less than the normal 1%”.

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Commercially available computers cannot generate truly random numbers using BIT logic (which is deterministic by nature as you said). However, computers CAN (to our knowledge) generate random numbers if they model some a process that IS truly random, which so far is almost always (always?) based on some physical phenomena. As a side note, quantum computers are trivially capable of generating random numbers through trivial means.

Realistically speaking though, DW does not generate random numbers for pulls through thermal decay nor do they have quantum computers. However, there are methods to distribute the deviation from true independence to as small a bound as you want for low computational power. For DW , it’s probably as large as they think they can legally get away with but small enough so that no player is going to be able to reverse engineer it, which is a very easy threshold to meet with modern tech.

There have been numerous cases of online casino algorithms in past decades being reverse engineered due to the devs using lazy pseudorandom generators, but that sort of thing is a relic.

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2 goes against the principle of RNG.

Computers’ randomness does not exist per se. If you go back to an old programming language like TurboPascal and create a program that will print 10 random numbers between X and Y, you will ALWAYS get the same sequence. Until you use a specific function designed to prevent that.
If you’ve played games that allow you to generate random maps (Worms, Minecraft, …), you might be familiar with the term ‘seed’, which is a sequence that will be used to base the randomness upon (so reusing the same seed will produce the exact same map each time).
The purpose of the specific function I mentioned is basically to create such a seed before starting to do random stuff. It is generally based upon time as it is considered to be unique each time you will decide to use it. Obviously, it’s now using something as precise as nanoseconds. But if it was using seconds only, you could again have identical sequences if each occurrence happened during the same second, thus the nanoseconds.
It is theoretically possible to predict it, but realistically impossible as it would take massive amounts of computing to tell you that you need your roll request to hit the server at a very specific time (down to the nanosecond), which is pretty much impossible to do because you can’t predict your internet’s responsiveness at that exact time, and any microscopic variation will be enough to make you miss your time target. Not to mention that you would also need specific and precise secret information about the server, from its uptime to stuff like various software versions to even start thinking about doing such calculations.

On the side of expected rates, it is only reflective of “give me a number between 1 and 100,000” where 1 to 700 equals “Rate Up SSR” (0.700%), 701 to 715 equals “Saber Artoria” (0.015%), and so on for each rollable card on that banner. (not necessarily in that order, but you get the jazz)
This is what happens for each card, always. You get a random number, and that number equals a specific card that you receive.
The ONLY tampering that happens is the guaranteed gold card in a 10-rolls (safe to assume all the SR servants/CEs are contained in a given range of those numbers and your guaranteed roll works the same but could be something like from 5,001 to 25,000 instead (if all 5% of SSR are the first 5,000 and the 20% SRs are following them in the list)

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I do not care because rate up is a lie. It might say .7, when when you have E luck, you have to convert the math to the proper .5.