How to get the expected running time of an algorithm
Usually you assume uniform distribution on inputs which share a certain parameter. 
You could assume uniform distribution on e.g.


All inputs of length $n$, e.g. if your input looks like $w_1\#\dots\#w_k$, where $w_i\in\{0,1\}^*$, $n=k-1+\sum_{i=1}^k|w_i|$.
All multisets (bags) of $n$ integers up to $k$
All $n$-tuples of integers integers up to $k$ (different,since e.g. $\{1,2\}_b$ corresponds to $(1,2)$ and $(2,1)$ and $\{1,1\}_b$ only to $(1,1)$)


You can combine this with an assumption about the distribution of the parameter, say the size $n\sim \mathrm{Geo}(0.5)$. However, this is unusual, most would consider the limit of probabilities as your parameter $n\rightarrow\infty$ instead.


If you choose 3, the probability of no repeated numbers can be calculated this way:
There are different $\binom{k}{n}$ sets of size $n$ and each set corresponds to $n!$ $n$-tuples. There is a total number of $k^n$ $n$-tuples, so
$$P(\text{no repeated number})=\frac{\binom{k}{n}\cdot n!}{k^n}$$