I had the same problem. Given a set where each item has a probability and whose items' probabilities sum up to one, I wanted to draw a series of items efficiently, i.e. without sorting anything and without repeatedly iterating over the set.
The following function draws the lowest of $N$ uniformly distributed random numbers within the interval $[a,1)$. Let $r$ be a random number from $[0,1)$.
\begin{equation} \text{next}(N, a) = 1 - (1 - a) \cdot \sqrt[N]{r} \end{equation}
You can use this function to draw an ascending series $(a_i)$ of $N$ uniformly distributed random numbers in [0,1). Here is an example with $N = 10$:
$a_1 = \text{next}(10, 0)$
$a_2 = \text{next}(9, a_1)$
$a_3 = \text{next}(8, a_2)$
$\dots$
$a_{10} = \text{next}(1, a_9)$
While drawing that ascending series $(a_i)$ of uniformly distributed numbers, iterate over the set of probabilities $P$ which represents your arbitraty (yet finite) distribution. Let $0 \leq k < |P|$ be the iterator and $p_k \in P$. After drawing $a_i$, increment $k$ zero or more times until $\sum p_0 \dots p_k > a_i$. Then add $p_k$ to your sample and move on with drawing $a_{i+1}$.
If you wonder about the $\text{next}$ function: It is the inverse of the probability that one of $N$ uniformly distributed random numbers lies within the interval $[a, x)$ with $x \leq 1$.