Suppose we have two bins A and B by iteratively distribute batches of m balls between those bins.
Let $|A|,|B| =$ the number of balls in A and B respectively at the beginning of a time step
Initially, $|B| = 0$ $|A| \geq 4$ (In the data $|A|$ would have tens of thousands)
Every step we take exactly m balls and with $$p_A = \frac{|A|}{|A|+|B|}, p_B = \frac{|B|}{|A|+|B|}$$
Add each ball to the selected bin, and add m balls to B.
How many balls should we expect to find in B after this process repeats n times?
I have been thinking of this as a tree, Because every step there are m ways the new balls can be arranged. So the tree branches m times at every branch but because of the number of ways to reach any given value for $|B|$. I suspect that the number of paths might be multinomial coefficients but I am well and truly stuck by that point.