In a "Statistics & Probability" course, the textbook (not in english) asks in one of its end-exercises, to provide a formula/expression to the following problem:
- In a circular table, $n$ kids are initially seated. Then, $m$ lego pieces are randomly distributed amongst them. The kids may play together and combine their legos if they are at most $k$ seats apart. An $L$-go is a piece consisting of at least $L$ pieces. What is the probability that an $L$-go was formed?
- For given $n,k,L$, what is the expected number of lego pieces ($=m$) on the table when an $L$-go is formed for the first time?
It's marked with a double asterisk (**) so it is supposed to be difficult. I tried to produce a recursion of some sort relying on base cases such as $k=0$ and $L=0,1$ or identify which sort of distribution this is, but I am actually clueless.