Studying the probability of an event Let $W:=(W_1,W_2,W_3,...,W_k)$ be a random vector of dimension $k \times 1$ taking values $(1,0,0,..,0)$, $(0,1,0,...,0)$,$(0,0,...,1)$. Assume that $W$ has a discrete uniform distribution. 
Let $1\{.\}$ be an indicator function taking value $1$ if the condition inside is satisfied and $0$ otherwise.
Let $r_1,r_2,...,r_k$ be some real scalars $\geq 0$. 
Let $\mathcal{G}:=\{j \in \{1,...,k\} \text{ s.t. } r_j >0 \}$, i.e. $\mathcal{G}$ is the set of the subscripts of the strictly positive numbers among $r_1,r_2,...,r_k$. 
Let $|\mathcal{G}|$ denote the cardinality of $\mathcal{G}$. 
Let $\gamma>0$.
How does the probability of
$$
\{ \sum_{j \in \mathcal{G}} r_j * 1\{W_j=1\}>\gamma\}
$$
vary with $|\mathcal{G}|$ if we fix $\gamma$ and $r_1+r_2+...+r_k$?
 A: Lets re-cast a few concepts to clarify the issues:
Based on your description, only $1$ component of $W$ can be non-zero. Therefore, 
$$\overrightarrow{W}\in \{0,1\}^k:\sum_{i=1}^k W_i = 1$$
$$r_i\geq 0 \implies \sum_{j \in \mathcal{G}} r_j * 1\{W_j=1\}=\overrightarrow{r}\cdot\overrightarrow{W}$$
Now, we are assuming:
$$\sum r_i=R>0$$
You are asking about the following probability:
$$G:=P(\overrightarrow{r}\cdot\overrightarrow{W} > \gamma|R,\gamma>0)$$
Specifically, you want to know:
Does the probability depend on $N:=|\mathcal{G}|$ (defined in your post) if we know the values of $\gamma$ and $K$?
Case 1: $\gamma \geq R \implies G=0$
Case 2: $\frac{\gamma}{N}<\frac{R}{N}\leq\gamma$
Now,  $N\leq k$ by definition, and we know that there exists some combination(s) of the $r_i$ whose sum exceed $\gamma$. Case 2 implies that not all the coefficients can be greater than $\gamma$.
Without knowing the specific values of $r_i$, the vector of $r_i$ could fall into any one of the three following categories:


*

*All $r_i \in \mathcal{G}$ are required to exceed $\gamma \implies G=0$

*Subsets of $r_i \in \mathcal{G}$ are required to exceed $\gamma\implies G=0$

*Up to $\min\left(\lfloor \frac{R}{\gamma}\rfloor,N\right)$ of the $r_i \in \mathcal{G}$ are individually greater than $\gamma\implies G\leq \frac{\min\left(\lfloor \frac{R}{\gamma}\rfloor,N\right)}{k}$


Case 3: $R>N\gamma$:
Here, we know we no longer need all  $r_i \in \mathcal{G}$, but we also can allow every single  $r_i \in \mathcal{G}$ to be greater than $\gamma$, and we know that at least $1$ of the coefficients is greater than $\gamma$.
Therefore $\frac{1}{k}\leq G \leq\frac{N}{k}$
So, the probability $G$ is partly controlled by $N$.
