First, let’s understand how we calculate the probability of getting a certain number of copies in one shop:
Pcell=(total champ / totalcost pool ) * Pcost
The probability to be offered a champion in one individual cell depends on the number of copies of that champion that are available, the number of copies of same-cost champions that are available, and the odds of getting a champion of that cost which depends on the level.
Equation (2)
Pshop (X = x ) = Pcellx (1-Pcell )5-x(5Cx )
We use the above binomial probability equation to determine the probability of being offered exactly x copies in a given shop.
We are assuming that each cell is calculated in parallel and each has the same probability to get a champion. This makes the calculation easier while only having a small effect on the result if the assumption turned out to be false.
When calculating the probability to be offered a champion in a certain number of rolls, r, it is less useful to search for an exact number of copies and much simpler and more realistic to search for at least a certain number of copies. After all, when needing one last copy to complete your team composition, a few extra copies doesn’t hurt and should be taken into account.
Equation (3)
Prolls ( X>=1, R = r) = 1 - Prolls(X < 1, R = r) = 1 - P rolls (X = 0, R = r)
Equation (4)
Prolls ( X>=2, R = r) = 1 - Prolls(X < 2, R = r) = 1 - P rolls (X = 0, R = r) - Prolls ( X = 1 , R = r)
Equation (5)
Prolls ( X>=3, R = r) = 1 - Prolls(X < 3, R = r) = 1 - P rolls (X = 0, R = r) - Prolls ( X = 1 , R = r) - P rolls (X = 2 , R = 2)
To calculate Prolls we once again use a binomial distribution and Pshop from equation (2):
Equation (6)
P rolls ( X = 0, R = r ) = Pshop (X = 0)r
Equation (7)
P rolls ( X = 1, R = r ) = Pshop (X = 1)1 (1 - Pshop ( X = 0))r - 1 (r C1)
Equation (8)
P rolls ( X = 2, R = r ) = Pshop ( X = 2 )1 (1 - Pshop ( X = 0))r - 1 (r C1) + Pshop ( X = 1 )2 ( X = 1 )2 (1 - Pshop ( X = 0))r - 2 (r C2)
Equation (6) contains an extra term because there are two different ways to be offered exactly two champions: having two in one roll and zero in the rest, and having one in two different shops and zero in the rest.
By combining equations equations (2) through (8) we arrive to our answer.