On the need for a self-correcting function in the PCP theorem Original proof of the PCP theorem, uses self-correction property of linear functions. 
Assume we have $f: \{0,1\}^n \rightarrow \{0,1\}$, a function or table of values,
that is $(1-\delta)$-close to some linear function $L$.
Using $f$ we would like to compute $L(x)$ for any desired $x$ with high accuracy. Such a
procedure is called a self-correction procedure since a small amount of errors in the table $f$ can be corrected using probes only to $f$ to provide access to a noise-free version of the linear function L.

Procedure $\text{SelfCorr}(f,x)$ is:


Select $y\in\{0,1\}^n$ uniformly at random.
Return $f(x+y)−f(y)$


One can prove if $f$ is $\delta$-close to a linear function $L$ for some $\delta&lt;1/4$, then for any $x\in\{0,1\}^n$ the above procedure $\text{SelfCorr}(f,x)$ computes $L(x)$ with probability at least $1−2\delta$.

My question is: Why we don't probe $f$ at $x$ directly. By this way, error of procedure is at most $\delta$ (that is better than $2\delta$ error of $\text{SelfCorr}(f,x)$)?
The probability of error when sampling $f(x)$ is $\delta$ when $x$ is chosen at random. Using self-correction, the probability of error is $2δ$ for all $x$ (not only random $x$)