Callisto's Appearance Rate Chart

[Callisto]

I've seen a few people from time to time ask things on Discord about summoning odds of certain heroes on different banners. Because of that, I thought I would go ahead and put my improved Appearance Rate Chart in a publicly accessible place. This chart will be updated with each new banner as soon as conveniently possible for me.

https://docs.google.com/spreadsheets/d/1o6VgA-f40K9Rc1xAtAf1JSgvuxdGGpy7MGPujR_9xnk/edit#gid=1084666508

The Yellow section of my chart is broken down by pity rate, so to find calculations relevant to your current situation, just line up the pity rate, and whatever row you're looking at. I'm sorry that my calculations do not go above 4%, I may one day expand it to 5%, but for the foreseeable future, to keep things small and concise, I won't be adding any higher pity rate calculations.

Here's a breakdown of what each different category means on my chart:

"Odds of summoning a focus unit on their corresponding color"

• This is the odds of summoning a focus unit on a single orb of their color. Example: With a 3.25% pity rate, the odds of one Green orb on the Sacred Memories banner containing Myrrh is 5.75%

"Odds of an Orb being"

• This area tells you the odds of an orb appearing as a certain color on your summoning session. Example: On the Sacred Memories banner, with a 3% pity rate, an orb has a 30.25% chance of being Red.

"5 Star Appearance Rate"

• This area tells you the odds of obtaining ANY 5* unit when summoning on the corresponding color. Example: On the Sacred Memories banner, when choosing a Red orb with a 3% pity rate, that one particular Red orb has a 6.84% chance of being ANY 5* unit
• The Areas "4 Star Appearance Rate" and "3 Star Appearance Rate" work in exactly the same way.

"Appearance Rate of Individual Heroes"

• This is one of the most complex areas of the sheet, but is also the most useful. This area allows you to figure out the summoning odds of any summonable unit in one pull of their particular color. To use this area of the sheet, first, pick the banner you plan on summoning on, then just pick a summonable unit, let's use Lilina for instance. To find the odds of Summoning Lilina, merely find the row that matches her color and rarity: "Red 5-4 Star", and then pick your pity rate; let's use 3.75%. According to the sheet, when summoning on the Sacred Memories Banner with a 3.75% pity rate, a Red Orb has a 2.04% chance of being Lilina.

The Blue Section of the sheet to the right works slightly different, it approximates your odds of pulling a particular focus unit in a particular number of orbs, when sniping for that unit. For Example, your odds of summoning L'Arachel on the Sacred Memories Banner with 200 orbs (approximately 40 pulls), is 76.31%

 

If any of you would like to see what goes on behind the scenes or would like to check my math, just let me know and I'll PM you a copy of the sheet. If you happen to find anything incorrect with my sheet, please tell me! Good Luck Summoning!

[eclipse]

I'm gonna move and pin this in the place where we have other data and whatnot.

[Lushen]

I'll also mention some very simple probability formulas that could be useful in conjunction with this chart.

If you want to know what your odds (%) of getting a unit in a particular amount of pulls (N), you can use this formula.
Odds of pulling at least one  :   (1 - (1 - %/100 ))^(N)

Keep in mind that your odds change every 5 pulls.  So to 'combine' the odds (in an OR fashion) you can simply add them.  So,
Odds of pulling in first 5 orbs + Odds of pulling in next 5 orbs + etc. etc.

By combining these two and the chart you can find out exactly what your chances are of getting a unit if you decide to spend all your orbs.

 

Thanks for the chart, I was just assuming it was 3% for simplicity before but this is better.  It's still not perfect b/c if you pull a different 5* your rates will reset which this wouldn't account for.

EDIT:  I just noticed the right side of the graphs do something very similar.  Out of curiosity do they account for resetting odds if you pull a different unit by mistake or not?  

Edited January 29, 2018 by Lushen

[Callisto]

22 minutes ago, Lushen said:

I'll also mention some very simple probability formulas that could be useful in conjunction with this chart.

If you want to know what your odds (%) of getting a unit in a particular amount of pulls (N), you can use this formula.
Odds of pulling at least one  :   (1 - (1 - %/100 ))^(N)

Keep in mind that your odds change every 5 pulls.  So to 'combine' the odds (in an OR fashion) you can simply add them.  So,
Odds of pulling in first 5 orbs + Odds of pulling in next 5 orbs + etc. etc.

By combining these two and the chart you can find out exactly what your chances are of getting a unit if you decide to spend all your orbs.

 

Thanks for the chart, I was just assuming it was 3% for simplicity before but this is better.  It's still not perfect b/c if you pull a different 5* your rates will reset which this wouldn't account for.

EDIT:  I just noticed the right side of the graphs do something very similar.  Out of curiosity do they account for resetting odds if you pull a different unit by mistake or not?  

The charts on the right do not account for resetting odds. 

I've done calculations on what the mean pity rate is over time (I found it to be around 3.57%), but that only holds true over truly massive amounts of orbs, not with smaller, more normal amounts of orbs, since with orb sessions of 200 or so, it's literally impossible to reach really high pity rates. For those formulas on the right, I just assume that the pity rate is 3% on every single pull, so those odds on the right slightly underestimate your chances (which I find more preferable to overestimation, which might get some people's hopes too high.)

I have considered doing calculations on what the mean pity rate is over 50 orb/100 orb/200 orb ect. sessions, but I do not think that doing that would change the odds by more than 1 or 2 percentage points, so I don't think it's worth it. Maybe some day I'll get the motivation to do it though, lol. 

Edited January 29, 2018 by Callisto

[Lushen]

1 minute ago, Callisto said:

The charts on the right do not account for resetting odds, the calculations could become quite complex when you think of every possible way that your pity rate can be reset. 

I've done calculations on what the mean pity rate is over time (I found it to be around 3.57%), but that only holds true over truly massive amounts of orbs, not with smaller, more normal amounts of orbs, since with orb sessions of 200 or so, it's literally impossible to reach really high pity rates. For those formulas on the right, I just assume that the pity rate is 3% on every single pull, so those odds on the right slightly underestimate your chances (which I find more preferable to overestimation, which might get some people's hopes too high.)

I have considered doing calculations on what the mean pity rate is over 50 orb/100 orb/200 orb ect. sessions, but I do not think that doing that would change the odds by more than 1 or 2 percentage points, so I don't think it's worth it. Maybe some day I'll get the motivation to do it though, lol. 

Ah interesting.  Yea I toyed around with the idea of making something that accounted for pity rate resets but decided against it.  Nevertheless this is the most accurate calculator I've seen so far.  I'm really big on calculating my odds before deciding to summon.  Thanks!

[Ice Dragon]

16 hours ago, Callisto said:

I have considered doing calculations on what the mean pity rate is over 50 orb/100 orb/200 orb ect. sessions, but I do not think that doing that would change the odds by more than 1 or 2 percentage points, so I don't think it's worth it. Maybe some day I'll get the motivation to do it though, lol. 

The mean rate for a 5-star focus character is pretty much never going to be more than 1 percentage point more than the base. The mean 5-star focus rate is pretty much always going to be close to 3.5% on a color with a focus character (because the mean number of pulls before a reset is around 12-20, and getting your 5-star on pull 20 gives you an average 5-star focus rate of 3.375% during that streak), which is only 1/6 of the base higher than base. Since the typical base focus 5-star rate when pulling that color is less than 6%, even for green, you'll pretty much never see an increase of more than 1 percentage point.

[Rezzy]

Nice chart!  It will be nice to know just how good or bad my luck is for a given focus.

[Arcanite]

Took you long enough, Cal!

[Lushen]

Thanks for getting it updated for the legendary so quickly.  Really eye opening to see how terrible the legendary banner is for everyone who isn't green.

[Callisto]

8 hours ago, Lushen said:

Thanks for getting it updated for the legendary so quickly.  Really eye opening to see how terrible the legendary banner is for everyone who isn't green.

Yeah, it's a shame the Legendary Banners are so lackluster for pulling individual heroes. They aren't so bad if you want all three characters of the same color, or just wanr 5 stars and dont care who they are, but otherwise they're just not worth pulling on.

[Nébulya]

This is really great. Thanks a lot for it! It's going to 

be really useful. 

Edited February 23, 2018 by Metakirby

[Lushen]

Is this chart updated given the 4* & 5* -> 3* & 4* changes?  

From this it looks like the gaps between the number of colors should be expected to decrease meaning that the percent changes should be slightly more consistant with color than before.  (at a glance it seems more reds got demoted than greens)

Spoiler

 

Edited April 11, 2018 by Lushen

[Callisto]

On 4/11/2018 at 5:55 PM, Lushen said:

Is this chart updated given the 4* & 5* -> 3* & 4* changes?  

From this it looks like the gaps between the number of colors should be expected to decrease meaning that the percent changes should be slightly more consistant with color than before.  (at a glance it seems more reds got demoted than greens)

  Hide contents

 

This chart has been updated to reflect the changes on new banners. I must state that from my comparisons, 5☆ focus rates in general were affected by less than .1% up or down depending on the focus units. Most of the time the effect was less than .05%, but for green sometimes it would exceed that. Essentially, the drops had a near insignificant affect on focus rates.

The biggest thing to note here is that old 3-4☆s are now harder to summon. 

Edited April 13, 2018 by Callisto

[Lushen]

2 hours ago, Callisto said:

This chart has been updated to reflect the changes on new banners. I must state that from my comparisons, 5☆ focus rates in general were affected by less than .1% up or down depending on the focus units. Most of the time the effect was less than .05%, but for green sometimes it would exceed that. Essentially, the drops had a near insignificant affect on focus rates.

The biggest thing to note here is that old 3-4☆s are now harder to summon. 

Awesome - thanks for keeping up with this.