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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