For the case where $k = 3$ and $a = [1, 2, 2]$: Among all four subsets (of values ${1,2}$), two are bad ($p_{bad} = \frac{2}{4}$). So we see $p_{bad}=\frac{1}{2} \neq \frac{1}{k}=\frac{1}{3}$
Also if we have an array with counts of elements $\in {5, 10}$ and $k = 10$, then probabilities are the same as for $k = 2$ and counts $\in {1, 2}$.
Suppose we are querying the whole array.
For the case where $k = 3$ and $a = [1, 2, 2]$: Among all four subsets (of values ${1,2}$), two are bad ($p_{bad} = \frac{2}{4}$). So we see $p_{bad}=\frac{1}{2} \neq \frac{1}{k}=\frac{1}{3}$
Also if we have an array with counts of elements $\in {5, 10}$ and $k = 10$, then probabilities are the same as for $k = 2$ and counts $\in {1, 2}$.