Lost mass problem

In the simplest form (going to A Turing), the problem can be formulated in the following way: we have an infinite system of boxes: $B_1,B_2,\dots$ and allocate particles between them one by one with the given probabilities $p_1,p_2,\dots$ What is the probability that the particle with the number $(n+1)$ will be in the empty box? How we can estimate the total probability of the occupied boxes? Etc.