The problem of choice and the optimal stopping rule for a sequence of random trials

Suppose we have to choose an element from a finite set $A$ which consist of $n$ elements. Let $A$ be ordered by quality. We regard our choice as successful if the selected element is one of the best $r$ elements of $A$. Let us enumerate the elements of $A$ in such order as we learn them. After learning at we know the comparative qualities of $a_1,a_2,\dots,a_t$ but we know nothing about the quality of the remaining $n-t$ elements of $A$. While learning $a_t$ we can either accept it (then the choice is made) or reject it (then it will be impossible to return to it). We describe the optimal policy which secures the greatest probability of the successful choice and describe its asymptotical behaviour as $n\to\infty$.