Before revision:  For probabilistic recurrences, our approach can derive tighter bounds than the  well-established methods of and for classical algorithmssuch as quick sort, quick select, and randomized diameter computation .
After revision:  For probabilistic recurrences, our approach can derive tighter bounds than the well-established methods on classical algorithms. Moreover, we show that our approach could derive bounds comparable to the optimal bound for quicksort, proposed by McDiarmid and Hayward .