Asymptotic probabilities in a sequential urn scheme related to the matchbox problem

Generalizing the classical Banach matchbox problem, we consider the process of removing two types of ‘items’ from a ‘pile’ with selection probabilities for the type of the next item to be removed depending on the current numbers of remaining items, and thus changing sequentially. Under various conditions on the probability pn1,n2 that the next removal will take away an item of type I, given that n1 and n2 are the current numbers of items of the two types, we derived asymptotic formulas (as the initial pile size tends to infinity) for the probability that the items of type I are completely removed first and for the number of items left. In some special cases we also obtain explicit results.