We want to calculate the probility of a Lotterydraw called Gacha in a game that give Skills for the character.
There are 3 different rarity for skills: rare, epic, legendary
there are 4 different classes
Class 1 has 4 legendary skills, 5 epic skills, 5 rares.
Class 2 has 4 legendary skills, 4 epic, 3 rares
Class 3: 4 legendary, 3 epic, 3 rares.
class 4: 4 legendary, 4 epic, 4 rares.
Basic rule
to use 1 pull it cost 1 Stellatie, which can be bought for 100 Dawnium.
Pity system:
every 10 pulls guarantess a Rare or better skills
every 30 pulls guarantess an Epic or better skills
every 70 pulls guarantess a legendary skills
The pity is resetes once you obtain a skill of the corresponding quality
we want to get skills for class 4 and there's another class pity
every 8 skill shards guarantee a skill shard of the highest class
every 8 rare skills guarantee a skill of the highest class
every 8 epic skills guarantee a skill of the highest class
every 4 legendary skills guarantee a skill of the highest class.
obtaining a highest-rank skill of the corresponding quality resets the counter.
shard conversion
any duplicate skills aquired will be converted as follows:
each duplicate rare skill is converted to 10 skill shards of that specific Skill
each duplicate epic skill is converted to 30 skill shards of that specific Skill
each duplicate legendary skill is converted to 90 skill shards of that specific Skill
droprates are as follow:
Legendary skill: Base Rate 1%, Overall Rate (inkl.Guarantee) 2.41%
Epic skill: Base Rate 7%, Overall Rate (inkl.Guarantee) 7.9%
rare skill: Base Rate 22%, Overall Rate (inkl.Guarantee) 24%
Skill shard (of a specific skill) Base Rate 70% Overall Rate (inkl. Guarantee) 65.69%, you will get 3 skill shards if pulled
About skill shards:
skill shards are used to upgrade skills further
rare and epic skills can be upgraded into legendary quality (it's still the same skill just a better quality)
legendary can be upgraded to divine and divine to immortal
legendary has 4 sub qualities: base, 1 star, 2 star, 3 star, once they hit 3 star, they can be upgraded into divine, each subrank needs 60 shards and another 60 to upgrade from legendary 3 star to divine.
Mythic goes from base to 5 star before upgraded to divine, they use 90 shards each to upgrade
for divine base to 7 star, then to immortal, 120 shards for upgrades.
epic skills base to 2 star 15 shards for upgrades.
each 50 pulls with stellatie gives us 1 Covenite, covenite can be used in a different gacha.
this gacha only gives skills and shards of the class 4.
overallrate inkl. guarantee are as follow:
legendary skill 2.41%
epic skill 7.9%
rare skill 24%
skill shards (3 each time) 65.69%
50 pulls with Covenite gives 1 exchange ticket, they can be used to get 30 skill shards of any skill


please calculate how many Stellatie we need if we want to
get any 1 Legendary of Class 4 to Immortal
get any 2 Legendary of Class 4 to Immortal
get any 1 legendary + 1 Epic Class 4 skill to immortal
for any 2 legendary +1 epic Class 4 skill to immortal
and for any 2 Legendary +2 epic Class 4 skill to immortal