Hey, Seeker? I have some summoning math questions.

His name is Canas. 👀

I'm encouraged to see people take an interest in this.

Let's start with how I calculate the number of orbs expected to find a unit. There are actually ways slightly better than mine, and I think Akariss probably uses them, but my figures are decent.

First we determine the appearance rate of all units. According to information at the bottom of the "details" page of every banner, all units of the same rarity (not colour) share an appearance rate. So for any stone, it's like the game first decides whether it's 3*, 4*, 5* off-focus, or 5* focus, and then from there it picks a random unit in that pool and shows you the colour. Therefore, the appearance rate of any character of a specific rarity is equal to:
(appearance rate of that rarity) / (number of units of that rarity)
Ex: appearance rate of B.Celica = 3% / 4 = 0.75%

Once we have this, we can figure out the appearance rate of stones of any colour by adding together the appearance rates of all characters of all rarities of that colour. So I would have to add the appearance rates of 3* Marth, 4* Marth, repeat for all 3*-4* characters, then 4* Soleil, 5* Soleil, repeat for all 4*-5* characters, 5* Ayra, repeat for all 5* exclusives, and then add any focus units. There is a shortcut, which goes like this: "the 4* pool has has 105 units, and 31 of the are red, so the probability of a 4* unit being red is 31/105." Then I can add the probability of a 3* character being red, a 4* character being red, a 5* off-focus being red, and a 5* focus being red.
I'm not sure if my counts of the units in each pool are dead on because I haven't checked since YT banner, but for CYL2, the percentages should be close to:
R: 30.1%
B: 26.7%
G: 18.2%
C: 24.9%

The probability of finding a specific focus unit on any pull of a given colour is equal to:
(appearance rate of that unit) / (appearance rate of stones of that colour)
Ex: the probability of finding B.Celica on any single red pull is equal to:
0.75% / 30.1% = 2.5%
If this part doesn't make sense, please ask.

As far as I know, every good calculationist (PhoenixMaster, Akariss, etc.) would agree with me up to this point.

From here on I take the "lazy" route because I haven't figured out how to make Excel account for pity rates. What I do is I say "ok, so if the probability is 2.5%, that's equal to 1 in 40. So the math says if you pull 40 reds, you can expect 1 B.Celica." Factually, we know that you can expect it but it won't necessarily happen! So what some people do is they flip it and say "how likely is it that any red stone DOESN'T give you B.Celica?" and there's a not-too-complicated way of figuring out what the probability is that you DON'T get B.Celica in a given number of red pulls. But all the figures I've been giving have used the direct route, just ask the question, "the chance of pulling my desired focus unit is 1 in how many?"

There's one more thing I can do, and the other calculationists will almost certainly agree with me on this part as well. I can predict how often you will be able to pull 1, 2, 3, 4, and 5 stones from a round based on which colours you want. If you want, I can get into the math of it; it involves the probability that any given stone will be a desired colour and the formula for combinations. But what's important is that I can use it to predict how often you're spending 5, 4, or 3 orbs on a pull, which I can then use to calculate an average of how many orbs you're spending per pull. For example, this is oversimplified but just to show how it works, if you want blues, and 4 rounds out of 5 you only get one blue stone and the other round you get two, then you spend 24 orbs per 6 summons for 4 orbs per summon. On the CYL2 banner, the numbers look more like this (again, my counts might be out of date so don't take them for exact figures):
R 4.73 orbs per summon
B 4.78
G 4.88
C 4.80
The things to note are that red is the lowest, green is the highest (most people know from experience that getting more than one green in a round is rare), and the variation across colours is very small. The difference between red and green is the biggest, 0.15, so you spend 1 less orb per 7-ish summons, which means you get an extra pull every 28-35 summons (because a summon usually costs 4-5 orbs). It won't make much difference when the difference in probability of finding the focus unit is something like 1 in 24 vs 1 in 40; you would need to pull 16 more reds than greens so that one extra pull will make only a very small difference in the long run. You'll start seeing numbers like 4.5 orbs per summon only if you pull multiple colours, and of course, if you pull all colours, it does go down to 4.

Once I know the number of orbs per summon, I can multiply that by the number of pulls within which I expect to find the focus unit. For example, I established that B.Celica is around 1 in 40.
40 x 4.73 = 190 orbs. I expect to find B.Celica in 40 summons costing 190 orbs.

What happens when you pull all colours is that the calculation suddenly becomes really simple. The answer to "what is the probability that a 4* unit is the colour I want" is simply 58%, the appearance rate for any 4* unit. At this point, we're treating all colours as equal, so the probability of any specific pull revealing a focus unit is 3%, which means we expect a focus unit 1 in 33 pulls. (33.3 repeating if you want to be precise, but in practice there's no such thing as 1/3 of a pull.) By comparison, the numbers I currently have for CYL2 banner are: 1 in 40 if you pull only reds, 1 in 36 if you pull only blues, 1 in 24 if you pull only greens, 1 in 33 if you pull only colourless. So yes, if you pull all four colours you do see more focus units than if you pull only reds.

The reason why I still don't go for the CYL2 banner is because I can probably get the heroes cheaper if I go for them 3-focus unit banners with 3%/3%. Assuming no colour sharing (as is the case with CYL2, i.e. all other things are indeed equal), I'm looking at 1 in 30 for B.Celica, 1 in 27 for B.Hector, 1 in 18 for B.Ephraim, and 1 in 25 for B.Veronica. (You can compare those to the CYL2 banner figures I gave; they assume the same 3*, 4*, and off-focus 5* pools.)

There's no getting around the fact that greens are generally less expensive to pull; and at the same time, even for greens, we should look for the best opportunities to pull. For example, M.Grima on the fighter skills banner was around 1 in 18. It's just that for reds, you will tend to pay a lot more severely if you don't get picky with the orb prices you're getting. For example, compare the 1 in 40 on CYL2 with the coming legendary banner. If you like 2/3 reds on the legendary banner, you'll get one around 1 pull in 23 (but keep in mind that if you only want one Swordhart, for example, once you get him your rate drops down to 1 desired 5* in 45 pulls because you don't want him anymore).

You are correct in that Celica has the same chance of appearing as any other focus unit. That would be true whether or not you pulled all colours; it's just that if you don't pull reds, you'll never know if she appeared or not. The fact has not changed that if you isolate the red stones, you will still have more non-focus reds than if you isolate any other colour.