# Does the PRG game allow for bad randomness picks?

In the game-based definition, we say that $$G: \{ 0, 1 \}^n \rightarrow \{ 0, 1 \}^{\ell(n)}$$ is a pseudorandom generator if For all ppt distinguishers $$D$$, there exists a negligible function $$\nu$$ such that: $$Pr[D( r) = 1] - Pr[D(G(s)) = 1 ] \leq \nu(n)$$ Where $$r \gets \{ 0, 1 \}^{\ell(n)}$$ and $$s \gets \{ 0, 1 \}^n$$ are chosen uniformly at random. Now, $$Range(G)\subset \{ 0, 1 \}^{\ell(n)}$$. So there's a possibility that $$r \in Range(G)$$ even if it's picked uniformly at random. Are we assuming that grabbing a "bad" $$r$$ is unlikely, or is the game implicitly saying the two cases are: $$r \in Range(G)$$ and $$r \notin Range(G)$$?

I could see if $$\ell(n) = 2n$$ then grabbing a bad $$r$$ would be unlikely, but the stretch only needs to be at least $$1$$, so if $$\ell(n)= n+1$$ a uniform at random $$r$$ would be in the range with probability $$2^n/2^{n+1} =1/2$$. In this case, always saying it's from $$G$$ seems like it would win the game $$3/4$$ of the time.