Yes, when you run an algorithm, often specified as a sequence of gates, you get different results according to some probability distribution.
Assuming we are talking about an error-free computation (this is where, in this field, we use the terminology "noise" which is distinct from the "noise" of "signal to noise ratio"), then most algorithms that you will currently find are designed in such a way that as the size of the input increases (and you therefore use more qubits), the success probability tends to 1 (increasing signal to noise ratio, to use your terminology). However, for a fixed input, you cannot always just throw more qubits at the problem and somehow magically expect to improve the probability of the correct outcome. (That said, some algorithms such as phase estimation do work like that.)
Instead, what you can do is exactly what you'd do with probabilistic classical computation: repeat the computation many times and perform a majority vote for the answer. You could do this in parallel, using more qubits, but you can also do it sequentially, and not use any extra qubits at all.