Understanding Property Testing with a toy example
The problem is, given a string of length $n$ over $\{0,1\}$, to distinguish the following two cases:


The string is entirely 0.
The string is $\epsilon$-far from being entirely 0, i.e., has at least an $\epsilon$-fraction of 1s.


The proposed algorithm tests $S = 2/\epsilon$ positions, and returns (1) if all positions were 0. If the original string consisted of 0s then this algorithm always returns (1) correctly, and otherwise it makes a mistake with probability $(1-\epsilon)^S \leq e^{-2}$.

We choose $S = 2/\epsilon$ in the first step so that the error probability is constant, in particular less than $1/3$. If you want a different error probability, you'll choose a different value of $S$.