Let $b_i, i=1,\ldots,m$ be real numbers.
Let $r_i, i=1,\ldots,m$ be random variables with $P(r_i=1)=P(r_i=-1)=1/2$.
Consider group $\Pi_m$ of all permutations of the set $\{1,\ldots,m\}$. On the group $\Pi_m$ consider the normalized counting measure $\mu_m(A)=\operatorname{card}(A)/m!$ for $A\subset \Pi_m$.
Let $f:\Pi_m\longrightarrow R$ as $f(\pi)=|\sum_{i=1}^{m/2} b_{\pi(i)}-\sum_{i=m/2+1}^{m} b_{\pi(i)}|$, where $ \pi(\cdot)\longleftrightarrow r_i= \begin{cases} 1,&\quad \text{if} \quad \pi(i)\leq \frac m2\\[4mm] -1,&\quad \text{if} \quad \pi(i)>\frac m2 \end{cases}. $
Is it possible to calculate $\mu_m(f=0)$?