I have an odd problem I am trying to solve and I feel like it should be solvable, but I am having now luck. I will try to boil the math down to a simple ball/urn description to not clutter the problem with irrelevant details.
I have 2 Urns with different colored balls in each:
Urn #1 - 50 Blue, 4 Red, 4 White and 2 Black balls.
Urn #2 - 50 Blue, 5 Red and 5 White.
Every selection for either Urn is done with replacements, so P() is identical for each selection.
Each trial X balls are selected from Urn #1, each Red Ball will add 1 to X, each White Ball will add 2 to X. Each Black Ball selected will add 1 to variable Y, which begins at Y = 1, and when Y = 5, the selections are made from Urn #2 instead of Urn #1. Trials will continue until all X selections are made, counting how many selections are made at Y=1,2,3,4,5.
Blue Balls selected do not effect X or Y, but are still counted as selections and are replaced.
X > 0, assume X = 5 or greater.
I am trying to calculate 2 things, the average length of X (average number of selections) and the average percentage of selections for Y=1,2,3,4,5 (% of selections at Y=1, 2 ,3 ...) .
Hope this makes sense and thanks in advance for any insight. If needed I can try and flesh out in vague places. My main issues revolves around the change in Urns and that change in selection P()s.