You have a set of playing cards with 15 unique cards in it and no more. You are given 5 cards per minute, with replacement. You need 20 cards of a specific kind to get a stamp of the card type. So, there are 15 stamps to earn.
let cards(t) = the number of cards earned at a given minute
What is the probability of having x specific stamps (20 cards or more of that kind to earn it's stamp) given cards(t) cards?
My rough work below
let cards(t) = the number of cards earned at a given time
P(getting a specific stamp card) = 1/num stamps to get
P(getting 19 or less cards of a specific stamp) = cumulative binomial distribution (19 successes, cards(t) draws, P(getting a specific stamp card))
P(unlocking a specific stamps (20 cards or more of that stamp) ) as = 1 - P(getting 19 or less cards of a specific stamp)
So from this we expand to unlocking x specific stamps:
P(getting 19 or less cards of a specific stamp * x stamps ) = cumulative binomial distribution (19 successes * x?, cards(t) draws, P(getting a specific stamp card) * x? )
! This part below definitely doesn't work because you can unlock 0..x-1 stamps and still not unlock x specific stamps. I'm not sure what type of summation or equation I should use to calculate this part
P(unlocking a x specific stamps (20 cards or more of each stamp) ) as = 1 - P(getting 19 or less cards of a specific stamp * x stamps )
let X = random variable with x=0..numStamps outcomes with P(unlocking a x specific stamps (20 cards or more of each stamp) ) probability
# stamps expected to be unlocked would be: E[X] = Sum x=1..numStamps( P(unlocking a x specific stamps (20 cards or more of each stamp) ) * x )
I'm having trouble with the minimums thresholds of the 'buckets' when I try to calculate the probability of unlocking x stamps.
P(unlocking a x specific stamps (20 cards or more of each stamp) ) = 1 - Sum y=0..x-1 ( P(unlocking a y specific stamps (20 cards or more of each stamp) )
^ something inductive like this might work but it may be more difficult than what this requires
The closed form math for this ended up being too hard, so I wrote a small script to simulate the draws instead and plotted the averages/distribution.
Someday I'd love to learn the math to get this working!