GenImp - Hard + soft pity

Ausdruck 9: "c" left parenthesis, "m" , right parenthesis equals Start sum from "r" equals 1 to 90, end sum, "p" Subscript, "R" , Baseline left parenthesis, "r" , right parenthesis left parenthesis, "r" minus "M" Subscript, "e" "a" "n" , Baseline , right parenthesis Superscript, "m" , Baseline

Ausdruck 10: "M" Subscript, "e" "a" "n" , Baseline equals Start sum from "r" equals 1 to 90, end sum, "p" Subscript, "R" , Baseline left parenthesis, "r" , right parenthesis "r"

equals

62.2 9 7 3 3 2 0 3 9 6

Ausdruck 11: "S" Subscript, "d" "e" "v" , Baseline equals StartRoot, "c" left parenthesis, 2 , right parenthesis , EndRoot

equals

24.3 1 2 2 6 3 5 3 2 8

Ausdruck 12: "S" Subscript, "k" "e" "w" , Baseline equals StartFraction, "c" left parenthesis, 3 , right parenthesis Over "S" cubed , EndFraction

equals

negative 1.2 4 2 6 8 8 9 1 5 1 6

Ausdruck 13: "K" Subscript, "u" "r" "t" , Baseline equals StartFraction, "c" left parenthesis, 4 , right parenthesis Over "S" to the 4th power , EndFraction

equals

3.0 4 1 1 7 6 4 6 1 5 1

Ausdruck 14: "P" Subscript, "C" "o" "n" "s" "o" "l" "i" "d" "a" "t" "e" "d" , Baseline equals StartFraction, 1 Over "M" Subscript, "e" "a" "n" , Baseline , EndFraction

equals

0.0 1 6 0 5 2 0 5 1 7 8 5 5

E(h+s pity) < E(h pity) < E(no pity) (round up) ≈ 63 < 70 < 167

P(no pity) < P(h pity) < P(h+s pity) (based on above approximations) ≈ 0.59% < 1.43% < 1.59%

Based on the h+s pity system, with 160 primogems per roll: E(# primogems per 5-star) = 63*160 = 10 080 Compared to the naive estimate of 90 rolls per 5-star, which is 90*160 = 14 400