An urn contains balls of different colours, which are randomly drawn one at a time. After each draw the number of balls in the urn with the same colour as the ball last drawn is changed. Special cases are sampling with and without replacement, and Pólya sampling. The drawing stops when a given number of colours has been drawn a given number of times. The number of times the different colours have been drawn is studied in this paper by imbedding the urn scheme in birth processes. Both exact and asymptotic results are obtained. In particular, waiting times for sampling with and without replacement and for Pólya sampling are considered.

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Yl, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.