It doesn't depend just on n since you aren't just searching through n available socks, you are searching k places with n_k socks in each place and with p_k conf

It doesn't depend just on n since you aren't just searching through n available socks, you are searching k places with n_k socks in each place and with p_k confidence that you have thoroughly searched each place. I can't think of a more general problem that this might correspond to.