Expected number of full bins after throwing balls uniformly randomly to bins that have limited capacity
Let us have $N$ bins with the same limited capacity ($N_{max}=C$), in the sense that if a ball is threw into a bin that already has $C$ balls in it, the ball is discarded. After throwing (uniform random allocation) $L$ balls, what is the expected number of bins that are full?
considerations:
- since uniform random allocations, after $L$ launches of balls we should have an uniform distribution of $L/N$ balls in each bin
- if $L<C$, no bin can be full (few balls to have the chance to have one bin full), so the expected number of "full" bins should be 0.
- if $L \in [C,C\cdot N ) $ it should be expected to have $\dfrac{L}{C}$ full bins, and $N-L/C$ bins remain free to host a new ball.
- if $L>=C\cdot N$ no bin can be filled any more,all the bins reached their capacity. Any attempts of a new launch of a ball is rejected.
This is an idea, is there an analytical closed form way to express that?