Feeling like you’re going through a patch of incredibly good (or incredibly bad) luck with Summons? There’s actually a way to mathematically identify exactly how lucky — or unlucky — you’ve been. This article is fairly technical, but please continue reading if you would like to learn more.

How the Summon Gate works

Every time you make a Summon at a Summon Gate, the Hero you receive is selected randomly based on the appearance rates configured for that particular Summon Gate. Players can always check these rates; the full list can be accessed by tapping the [i] icon in the top left corner of a Summon Gate.

You will see that different probabilities have been set for different groups of Heroes. Appearance rates are lower for Heroes that are meant to be rarer. As Legendary Heroes are meant to be very rare, their appearance rates are lower than that of Epic or Rare Heroes.

It is VERY important to remember that each Summon is independent of each other — the outcome of previous Summons do not influence the outcome of future Summons in any way. Appearance rates always remain the same regardless of how many Summons you conduct.

The easiest way to understand this is to think about how coin flips work in real life! Even though coin flips are 50/50, you are NOT guaranteed to get 1 heads and 1 tails every time you flip a coin twice, because EACH flip has a 50% chance for heads and a 50% chance for tails. Similarly, a player is NOT guaranteed to receive a Hero of the Month (1.3% appearance rate) after 77 Summons, even if 77 times 1.3 is 100.1.

How this plays out in practice

For the purpose of this article, let us assume that you want to obtain the current Hero of the Month. (Please note that these calculations can be made for ANY hypothetical target Hero.)

In such a scenario, if you make a Summon and succeed in obtaining the Hero of the Month, that would be GOOD; if you fail to obtain the Hero of the Month, that would be BAD. So there would be only two possible outcomes from a Summon: GOOD and BAD.

In situations where there are only two possible outcomes AND probabilities remain constant and non-cumulative, a binomial distribution curve is guaranteed to emerge as the sample size expands. You may have seen similar bell-shaped curves before.

All players who make Summons will fall somewhere on this kind of curve. Notice that in a binomial distribution curve, distribution is symmetrical and has no skew; most data points cluster around the center, and the two tail ends taper off.

Now, using binomial distribution, let us calculate the probability of receiving a certain number of Heroes of the Month from 1,000 Summons. The appearance rate for the Hero of the Month is 1.3%, so one may expect to receive exactly 13 Heroes of the Month from 1,000 Summons. You will see, however, that what actually happens is a bit different.

Luck goes both ways

Contrary to what a layman might expect, binomial distribution tells us that only 11.07% of players who conduct 1,000 Summons will receive exactly 13 Heroes of the Month. 46.24% of players will receive less than 13 — but on the other hand, a whopping 42.69% of players will receive more than 13.

It’s clear that luck goes both ways — and although the chances of it happening are rare, luck can occasionally go quite far, too.

The above binomial distribution indicates that around 2.5% of players would receive more than 20 Heroes of the Month from 1,000 Summons. We could say that these players are luckier than roughly 97.5% of their peers. They might feel like they’ve entered a cheat code or something, but this is just how probabilities work.

On the other end, around 2.5% of players would, unfortunately, receive less than 7 Heroes of the Month from 1,000 Summons. These players would thus be unluckier than roughly 97.5% of their peers. These players may feel like there’s something wrong with the game, but again, this is just how probabilities work.

We definitely consider our system fair, as the same odds always apply to everyone equally. However, we also acknowledge that obtaining a desired Legendary Hero from a Summon Gate is not easy, especially when luck is not on your side — which is a big part of why the Soul Exchange is so popular amongst our players.

But hey, there's good news: with our Version 52 Update, we introduced the Fated Summon — a new quality-of-life feature that ensures players are rewarded with at least one free Legendary Hero after making a certain number of Hero Summons! Now you're guaranteed to always get at least some mileage out of each and every Summon you make.

Thanks for reading! Best of luck in the battles — and in all your future Summons.