Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous drawn letters and that $\sum_{n=1}^K p_n =1$; this model should be known in the literature as Bernoulli scheme.

Suppose moreover that the letters are not chosen uniformly at random, i.e. there exists a letter (without loss of generality the first one) whose corresponding probability $p_1$ satisfies the property $p_1 \geq p_n$ for all $2 \leq n \leq K$ and this inequality is strict for at least one letter.

We keep sampling letters according to this scheme until we create a string $s$ where all $K$ letters appear at least once.

How many occurrences of the first letter do one have on average in such string? If an exact result is impossible, can one bound this mean or say something its magnitude?

Any hint, partial solution or reference for this problem would be really appreciated.