Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are fixed values. Let $\beta$ be a field element picked uniformly random. Assume we have $\vec{r}=[r_1,..r_n]$, where $r_i\neq0, r_i \in \mathbb{Z}_p$. A subset (but not all) of $r_i$ can be 1. For those $r_i$ that are not 1 we have $r_i\neq r_j$. The $r_i$ values are NOT picked uniformly at random.
We define $T(x)=(x-\beta)\cdot g(x)$. So $T(x)$ is product of two polynomials $x-\beta$ and $g(x)$. Degree of $T(x)$ is fixed $n-1$, and the polynomials are defined over the field. We evaluate $T(x)$ at the elements of $\vec{x}$. This yields vector $\vec{y}=[y_1,...,y_n]$ where $T(x_i)=y_i$
Given $(x_1,y_1\cdot r_1),...,(x_n,y_n\cdot r_n)$ we interpolate polynomial $T'$, and we assume $T'$ has root $\beta$.
We shuffle the elements in $\vec{r}$ so an element in $i^{th}$ position is moved to a uniformly random $s^{th}_i$ position: $r_i \rightarrow r_{s_i}$
Given $(x_1,y_1\cdot r_{s_1}),...,(x_n,y_n\cdot r_{s_n})$ we interpolate polynomial $T''$.
Question: What is the probability that $T''$ has root $\beta$ ?