Probability of completing multiples of objects

Hey Guys!

I'm struggling with a posed question about completing a set of objects and would really appreciate some direction.

EX: There are 10 objects, each with an equal chance of appearing (10%), what is the chance that after obtaining 200 random objects, that you have completed a set of 4 of each object, given you can continue to gain objects after 4.

I started off using a binomial distribution (excel: BINOM.DIST.RANGE(N,10%,5,N)) to determine the chance at each object gained that the object does not contribute to the goal to use in estimation. (This is much easier completing a set of 1 each :( )

I get tripped up trying to tack on the chance of gaining a 5th object for a second set; where I think I end up with some overlap in probabilities. It would seem I need to individually factor in the chance of getting 5 of set 1 and then 5 of set 2 with the remaining objects, and also getting 6 of set 1, and then 5 of set 2, and so on and so forth, for every object gained through all 10 sets of 4. And I'm losing it here.

I feel as though there must be a formula or something I'm unaware of and would really appreciate being pointed in the right direction.



TS Contributor
this would still be quite involved mathematically. I just ran a quick script in R with 1 million sampes and the chance to get a sample of 200 parts where not all parts were sampled at least 4 times was about 1 in 10000 (1e-4).

I hope this helps