FEH Legendary Hero Fest Banner Statistic Analysis

The legendary Hero fest banner will arrive today at 11 pm PST. Are you excited and prepared?

We have talked about what the hero fest banner pulling strategy from hero strength perspective.  It’s a no brainer to draw from the banner if you need all 3 heroes of a color. But I bet some of you still wonder:

  1. I have some of the heroes already, shall I draw with all my orbs or wait for a normal 3% pool?
  2. I have only 80 orbs, but I want Hector, I want Brave Lyn, Ayra is nice to have, I don’t mind getting another Spring Camilla … , how shall I draw?

So this time, let’s see the questions from the statistic perspective.

The Motivation

Someone may ask, the probability is 8%, if you want one out of the three heroes, it’s 2.67%. And a normal banner has 3% up rate. Isn’t that clear enough? The answer is, this banner is special that it excludes all other 5 stars.

First of all, let’s make it clear that even though IS used 8% to generate 5 stars, the rate of when you see a certain color, and then open it, expecting to see a 5 star is totally different! Let me give you an extreme example. If you exclude all red 4* and 3*. So the only reds in the banner would be the 3 up heroes (Ike, Ayra and Celica). IS can still make them appear at 8% rate, but if you only open red stone (if you see red), the 5* rate is 100%. Of course the probability of seeing a red stone is low in this case, that’s why I wrote a simulator to replicate exactly what IS described they do to find out the real numbers.

The set up

I basically wrote a FEH Gotcha simulator which I call it FM FEh Simulator (FM is short for First Minute). I can specify who are the up heroes. what’s the initial up rate, 5 star rate, 4 star rate. My simulator handles pity rate, that does exactly the same as IS describes in the Gotcha banner, “More info”. That is, if you don’t have any 5* in 5 draws, the rate of 5 star will increase by 0.5%, distributed between the up and other 5* heroes. I have also handled the case if there are none of your preferred color. You have to draw at least one. And of course I know the first draw each round of 5 stones costs 5 orbs, last costs 3 orbs, while the rest costs 4.

And then I specify a way to draw from the banner, and let it run, say 10,000 draws. It will give me the result of what up heroes I got, how many, total 5 stars, spent orbs, and calculate the orbs needed per 5 star heroes.

The Data

I used real data in my simulator. Gotcha games in Japan are required to publicize the drawing mechanic. The rate you use, the rewards you give. So IS listed clearly what heroes are in what categories. I checked the most recent banner  – Tempest trial (featuring 4 brave heroes), and the oldest banner available today – Halloween which was available a month ago. They have exactly identical hero lists besides up heroes. So I put them into my database.

The methods

I used two different ways to simulate so that I can cross verify the results. One is based on Excel, with real hero data even for three stars. The other is programmed in Java, using only stats, since I realized I don’t care about what 4 star or 3 star heroes I got. I did keep the up hero list just for fun.

The good news is, results from the two simulators agree with each other.

The Result

In this 8% hero fest banner, if you try to draw only from one color, the orbs needed for each 5 star is:

Red 60
Blue 53
Green 45
White 53

 

Amazing, huh? I’ve never expected the difficulty to draw a hero of a certain color would be different! Green is the easiest and red is the hardest. I.e. you need more orbs to draw an up hero in red. No wonder my armor sub account obtained 10 Amelia with only 300 orbs but I could never do that for Ike even though Ike was in starter banner at 5% rate!

But on another thought it’s easy to understand. There are more red heroes in this game. So after it’s determined the stone is not an up hero, there’s a higher chance that it’s red. As a player who see the 5 presented stones, red stones appear more. But if you poll the red stones, there’s less chance for it to be an up hero compared with other colors. 1 out of 100 is smaller than 1 out of 20.

If you determine to draw from 2 colors only, or all 4 colors, the rate (orb / 5 star) is different.

Green and White 46
Green and White and Red 48
All 4 Colors 46

You see, the number of 5 stars you obtain is roughly the same if you draw from all 4 colors vs. only from green. So if you want more Hector, and don’t have enough orbs, feel free to draw only green.

If you have many orbs, it’s still recommended that you draw from this banner on those you need. If you need the hero, you will need to draw him or her anyway. Rather than wait until a 3% banner, you’d probably do it now. This is the orbs need to draw an up hero from a normal banner with 3% up rate:

Red 95
Green 76
White and Green 73

You see, if you need all 3, the rate in the hero fest banner is much better. You save 40% of orbs. If you need 2 out of the 3 in a certain color, it’s still better (obviously). If it’s 1 out of 3 that you are interested in, it’s not much worse. I have that simulation run for you too, but for red only. After all, who would *not* draw the green if you want the heroes!

2 heroes green 63
2 heroes white and green 63
2 heroes 4 colors 62
1 hero green 100
1 hero 4 colors 94

Btw my armored account has done 2 out of the 4 Michalis grand hero quests. I’m yet to beat the Inferno difficulty. I wish I had a strong red sword horseman!