I play gacha games, and recently I've wanted to compare the rates of the one that I play compared to competitors in the market to warn some friends of mine from starting one with rates that I feel are extremely bad.
Because others may not be familiar with gacha games, I felt the best analogy was an unfair die.
Assume there is a 20-sided, unfair die. The probability of rolling a 1 is 4%, rolling a 3 is 2.5%, rolling a 7 is 1%.
I am trying to find the probability of rolling at least one 1, one 3, and one 7, out of some number of die rolls.
I unfortunately have basically zero background in statistics.
If it were just one specific result I wanted, from my Googling, I could just use a standard binomial probability formula from any graphing calculator. Needing "at least" one success, say, for landing on a 3 at least once in 10 rolls means I could just do $1−binomcdf(10,0.025,0)$
But once I need multiple specific results, I have no idea what to do. Say if I wanted to land on 1, 3, and 7 at least once in the same 50 rolls.
I initially thought I could just subtract the probability of getting none of the 3, finding the probability I want by doing $1-binomcdf(50, 0.925, 50)$
But then I realized that once one desired result was acquired, any further instances of that result would actually constitute a failure, as it would not be fulfilling the "at least once" of the other two necessary results, so the probability of a failure roll would be changing through the trial.
How should I approach this issue?