Rolling The Dice: Gacha Probabilities

What Are Our Chances of Winning a 5-Star Ranger?

We all take a chance with our rubies hoping to play the GACHA machine for great rangers like KSM, Sorcerer Cony, etc. The vast majority of us have experienced our fair share of disappointment with this random generator. I read an article today where the author, 楊濟安, has taken the effort to crunch the numbers on exactly what our odds are of landing a 4 or 5 star ranger. His calculations are based on a healthy sampling from a couple hundred volunteers who shared their Gacha results. The full article goes into detail so if your Chinese (and curiosity of probability) is solid, do go and read the whole thing. Again, full credit goes to the author of the article, but I will pull a few details to discuss here for the benefit of our English speaking community. It's a burning question that we all want to know about because some of us (many of us) spend cold, hard cash and we would like to know just how much of a gamble it is.

• Full article in Chinese
• Data samples are here (courtesy of the Facebook group LINE Rangers 台灣社團)

So if you're not interested in all the gritty details (TL;DR and all that), during the "Double Chance on 5 Star" event, the probability of getting a 5-star ranger  is approximately 20%. The probability for 4-star is approximately 30% and 3 star is about 49-50%, based upon the samples obtained. Based on this, the probability of 5-star rangers goes down to only 10% when the event is not taking place. 

Other Interesting Findings

One of the assumptions that he takes on is that every turn of the Gacha is an independent event. This means that one turn of the Gacha has no bearing on the result of the next turn of the Gacha. He believes that the 6 results from 100 rubies Gacha are independent events because a lot of people reported getting four 4-stars and one 5-star, or three 5-stars and three 4-stars. This kind of result would not be possible if the game developer had set the 100 rubies Gacha to be dependent events. For instance, if we believe the Gacha is set up to predetermined dependent results, then we should often get results of three 3-stars, two 4-stars and one 5-star, with some variability but not to the extreme of some people getting four or five 4-star rangers in one shot of 100 rubies. I buy his line of reasoning for this assumption, and it is an important one to make.

The author imparts us with a bit of basic probability knowledge which may affect the way we interpret the possible results we get when we play. Suppose there were 10 marbles in a can and one of the marbles would win you a prize. 10 people line up to pick out one marble each, one at a time. Each time someone picks a marble, they put the marble back. The chances of each person winning the prize is 1 out of 10 -- easy, of course. Suppose the 10 people in line picked out a marble each and did not put their marble back after selecting. Now who has the highest probability of winning the prize? The answer is they all still have the same chance of winning, which is 1 out of 10. (The first person has a 1/10 chance he has picked the winning marble. The second person has a 9/10 chance that the first person didn't get the winning marble * 1/10 chance that he selected the winning marble = 1/10). All this is to say that the misconception that if you got something good from the Gacha, the chances of you getting something good again the next turn is lessened, is completely mathematically false. If the chance of getting a 5 star ranger is set by the game developer at 10%, then your chances each and every time you play the Gacha is 10%, no matter what you got in the past, no matter what someone you know recently got, no matter night or day or where you live. The numbers don't lie :)

Six 3-star rangers? Sad day.

Another question he answered was regarding the poor players who get six 3-star rangers with 100 rubies. How is this possible in a world where the chance to get a 3-star ranger one time is 50%? He uses normal probability distributions to determine that there is a 81.33% probability of not obtaining six 3-stars. If you happened to be the chap who got six 3-stars, you're in the unlucky 18.67% lot. It's not an insignificant chance if you ask me!

One last note is, he made some assumptions to show that playing 100 rubies at a time is only slightly better than playing 20 rubies at a time but the details are a bit fuzzy to me (lost in translation; I couldn't quite understand the assumptions he made to arrive at his numbers). All I can say is that he concluded that the overall result of doing 20 rubies 5 times versus 100 rubies 1 time, that the 100 rubies wins out, if only by a little. 

The article was really enlightening and fascinating. Might takes maybe a bit of knowledge in statistics and a lot of Chinese to understand, but totally awesome that someone took the time to not only crunch numbers but also explain all this stuff. Bravo and a big thank you to 楊濟安 and the people who shared their results to make this study possible. In the words of Bill Nye the Science Guy, "now you know!"

I know what you're thinking! You're thinking, 20%? Pfffltt. Screw that, I didn't get anything from my rubies! I got Lucas, Hunt, Vlad, Ryan and Jason over and over! Remember that's 20% overall in a relatively large sample pool. If you played the Gacha 100 times in a row, 20 rubies at a time, surely close to 20 of them will be 5-stars. 

I'll give you an example. I found out from my Macho Moon Friendship Point Event study that the chance to get Macho Moon during this event was in the neighborhood of 4%. However as it so happens on day one of the event, I played the Friendship Gacha 172 times before I got my very first Macho Moon. That translates to 0.6% in a sample of 172. Not even one percent! It seemed absurd to me at the time. But later on in the day as I continued to play, oddly enough, I got Macho Moon 2 out of 3 spins, which is 66.6% looking at a sample of 3. On day one I played the Gacha a total of 244 times and landed myself 10 Macho Moons. So it evened itself out to 4.1%. Day two and day three were not nearly as dramatic because Moon popped out much more evenly. But throughout the course of the day I did find myself inexplicably caught in chains of 1 and 2 stars, then later on, in chains of 3-stars. However, even the 1, 2 and 3 star quantities leveled themselves out if I played enough times in a day. This is just how statistics and probability goes. We arrive at a number that is the result of calculations from an extremely large amount of trials. Your reality after playing a handful of times may vary quite a lot from the average.

And let's not forget that the 20% is not the chance that you will get something useful to you. The 20% emcompasses the chances that you will get duds like Luke (who nobody wants lol), Xan, Sol and Alice (who are available from stage clear rolls). The chances of you getting what you want (like KSM, Dancer King Boss, Sorcerer Cony and the like), is actually quite a bit smaller. So let's not confuse what you want with the total probability of 5-star rangers. 

[Addendum: I emailed the company and asked them whether the probability of all rangers within the same star grade is the same throughout. I have been told that the probability of all the rangers is preset. They do discriminate among the rangers as they deem some rarer than others, and the rarer ones have a lower probability than the others. Therefore, while the general odds of getting 5-star rangers on a normal day may be 10%, within the 10% there is another variable which determines which 5-star ranger you'll get. We can safely assume that the more desirable rangers are the "rarer" ones they referenced. However, a smaller chance does not mean no chance, it is just less probable. It's like they say in the lottery business: You have to play to win. And if you don't ever Gacha, then you will never win. If you don't want to gamble, that's your decision. So that's it, I'll leave it at that!]

Hope this post was helpful to you guys out there! Happy gaming and good luck!