Wish/Expanded Wish Probabilities

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Mathematical Treatment

This is a list of tables that provides detailed breakdowns of the rates for obtaining items within a given rarity tier. For example, on the standard wish, there are 36 total items in the 4★ rarity pool, and because, just by chance, it is equiprobable to obtain any given item from the 4★ pool, we can calculate the rate of obtaining any given specific 4★ item whenever a 4★ item appears (the rate within the same rarity) using the following formula:

${\textstyle f(p,q)=\left({\frac {p}{p+q}}\right)=\left({\frac {1}{1+35}}\right)={\frac {1}{36}}\backsimeq 0.0277{\overline {7}}\left(2.78\%\right)}$

where $p$ is the number of times our outcome occurs within an equiprobable set, and $q$ is the number of times our outcome does not occur. If we want to know the probability within the general pool (i.e. how likely we are to obtain any given 4★ item from any given wish) we can multiply this rate by the base rate for 4★ items (5.1%), giving us a final overall rate of 0.14%. However, you may notice that this page gives a different rate (0.36%). This is because the rate that was used to multiply the initial "same tier" rate was not the base rate for 4★ items, but rather the consolidated rate, including guarantees. If we revise our calculations to use this higher number, 13%, we se that the outcome is now 0.36%, as expected.

This means that the rates on this page are only relevant when applying the law of large numbers. Over a small number of wishes, stopping rules and other end effects will have a significant impact on the actual apparent rate, and it will always be lower than the consolidated rate, and it will usually, but not always, be higher than the base rate.

On 5★ and 4★ Drop Probability

Promotional Character Banner

	With Pity	No Pity
5★ Featured Character	100%	50%
4★ Featured Character	100%	50%
4★ Specific Featured Character	33.33%	16.67%

Promotional Weapon Banner

	With Pity	No Pity
5★ Featured Weapon	100%	75%
5★ Specific Featured Weapon	50%	37.5%
4★ Featured Weapon	100%	75%
4★ Specific Featured Weapon	20%	15%

4★ Non-Featured Drop Probability

Although obtaining a non-featured 4★ item is supposedly 50/50 between characters and weapons, this isn't the case. The character banner's non-featured 4★ items are heavily towards 4★ weapons while conversely in the weapon banner's non-featured 4★ items are heavily towards 4★ characters. Counterintuitively, obtaining characters that aren't rate ups in the character banner is unlikely in the same way that obtaining weapons that aren't rate ups in the weapon banner is unlikely.

While an argument can be made against the false information given by the Wish details, the advantage of the mechanic is it ensures a sort of balance between characters and weapons such that the player cannot have too many of the other, regardless of the featured banner being wished.

Non-Featured Drop Probability

Banner	Non-Featured Characters	Non-Featured Weapons
Character Banner	15%	85%
Weapon Banner	92.5%	7.5%

Overall 4★ Item Drop Probability

Banner	Featured Items	Non-Featured Characters	Non-Featured Weapons
Character Banner	66.67%	5%	28.33%
Weapon Banner	80%	18.5%	1.5%

Epitomized Path

A new mechanic introduced in the Weapon Event Banner now ensures the chosen featured weapon to be guaranteed by the third win of a 5* weapon. This creates an upper bound of 240 wishes (3 5* weapon wins) for the obtaining the chosen featured weapon. In addition, the chance of obtaining the chosen weapon within n 5* weapons wins is in the table that follows. This puts the consolidated probability of the chosen weapon to be 50.39% of all 5* weapon wins or an while the other featured 5* weapon and non-featured weapons being 32.28% and 17.32%, respectively.

No. of 5* Weapon Wins	Probability	Cumulative
1 win	37.50%	37.50%
2 wins	26.56%	64.06%
3 wins	35.94%	100.00%