Let $X, Z_{1}$, and $Z_{2}$ be independent random variables taking values in the set $\{0,1\}$. $X$ is uniformly distributed in $\{0,1\}$, while the distributions of $Z_{1}$ and $Z_{2}$ are such that if we define $Y_{1}=X+Z_{1}$ and $Y_{2}=X+Z_{2}$, where addition is modulo $2$, then
\[\begin{array}{l}\operatorname{Pr}\left(Y_{1}=1 \mid X=0\right)=\operatorname{Pr}\left(Y_{1}=0 \mid X=1\right)=p_{1}, \text { and } \\ \operatorname{Pr}\left(Y_{2}=1 \mid X=0\right)=\operatorname{Pr}\left(Y_{2}=0 \mid X=1\right)=p_{2} .
\end{array}\]
Consider the optimal estimator of $X$ from observations of $Y_{1}$ and $Y_{2}$, defined by the following optimization problem:
\[\min _{f(\dots)} \operatorname{Pr}\left(X \neq f\left(Y_{1}, Y_{2}\right)\right),\]
where the minimization is over all functions $f$ which map a pair of observation bits to an estimate bit. What is the value of the above minimum? You can assume that $p_{1}, p_{2} \leq 1 / 2$.
- $\max \left(p_{1}, p_{2}\right)$
- $\min \left(p_{1}, p_{2}\right)$
- $\left(1 / p_{1}+1 / p_{2}\right)^{-1}$
- $\left(1+1 / p_{1}+1 / p_{2}\right)^{-1}$
- None of the above