On the estimation of efficiency of voting procedures

We consider the problem of a collegial decision on the basis of individual opinions of $n$ experts deciding independently, the probability of a correct decision by the $i$th expert being equal to $p_i$, where $\frac12<m\le p_i\le M<1$, $i = 1, 2, \ldots , n$. It is shown that, for the error probability of the optimal collegial decision, the estimates
$$ \frac{1-M}M\binom{n}{[n/2]}\prod_{i=1}^n\sqrt{p_i(1-p_i)}\le\mathsf{P}^{\mathrm{err}}_{\mathrm{opt}}\le\frac m{2m-1}\binom{n}{[n/2]}\prod_{i=1}^n\sqrt{p_i(1-p_i)}. $$
are valid.