So the general story is "we draw identical marbles (with replacement) and place them randomly in 3 different bins" (generic stars / bars), we continue drawing until a bin has exactly 10 marbles.
- I would like to find the distribution.
I assumed that finding the total number of combinations would be easier if I partition the space by total number of marbles drawn {10,11,...,28} and then count the combinations for each. However the generic stars / bars solution doesn't apply since the restriction on each bin is 0=<Bi<=10 and least one bin has exactly 10. I looked at various solutions involving restrictions and generatic functions but can't quite find the right one.
In addition to the above restrictions every time we draw a marble there is a 1/5 chance to draw 2 marbles instead and place them in one bin.
There is 1/10 chance to draw 3 marbles and place each in a different bin, this is in addition to #2, ie. #2 can happen, #3 can and 2 and 3 can ie. 2/100 chance to draw 6 balls and place 2 in each bin.
(Edit) Some clarifications:
A: I'm looking for the chance that we drew exactly 10,11... or 28 times for the base case, eg. there are exactly 3 ways to have drawn 10 times. So the order in which the bins are filled does matter here.
B: As to #2, #3 as I mentioned in A the order in which the bins are filled matters since I'm looking at the distribution of the various sums. The bins overflow (ie. if you have 9 and place 2 you still have 10), and yes with #3 it's possible to have 10 marbles in all 3 bins (in the base case however that would be impossible since process stops when we reach 10 in any).
I guess an alternative way to describe the base case would be we have a 3 sided die (A,B,C), we roll a die until we have either 10 A's or 10 B's or 10 C's, what is the chance to have rolled 10 times, 11 times etc.
for #2 and #3, the die would be weighted with each side representing the combinations (we get 1 A, or we get 2 B's or we get ABC etc).