$M=eVt/h$ is the average number of injected particles. The joint probability distribution for transferring $Q_{1\sigma}$ spin-$\sigma$ electrons in lead 1, $Q_{2\sigma}$ spin-$\sigma$ electrons in lead 2, etc. is related to the characteristic function by the relation (we assume that no polarizers are present): $$P(Q_{1\uparrow},Q_{1\downarrow},Q_{2\uparrow},\ldots)=\frac{1}{(2\pi)^{2n}}
\int_{-\pi}^{+\pi} d\lambda_{1\uparrow} d\lambda_{1\downarrow}
d\lambda_{2\uparrow}
\ldots ~\chi(
\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}}
) ~e^{i\vec{\lambda_{\uparrow}}\cdot\vec{Q_{\uparrow}}}
~e^{i\vec{\lambda_{\downarrow}}\cdot\vec{Q_{\downarrow}}} ~.
\label{counting}$$

In the rest of the paper we will consider systems where only two counting terminals are present. In particular, while the counting terminals are kept at the lowest chemical potential, all other terminals are biased at chemical potential $eV$. For later convenience, we write down the most general expression for the characteristic function when spin-$\sigma$ electrons are counted in lead 1 and spin-$\sigma'$ electrons are counted in lead 2: $$\begin{aligned}
\chi_E (\lambda_{1\sigma},\lambda_{2\sigma'})=1+
\left( e^{-i\lambda_{1\sigma}} -1 \right) \langle \hat{N}_{\text{O}}^{1\sigma} \rangle +
\left( e^{-i\lambda_{2\sigma'}} -1 \right) \langle \hat{N}_{\text{O}}^{2\sigma'} \rangle +
\nonumber\\ +
\left( e^{-i\lambda_{1\sigma}} -1 \right) \left( e^{-i\lambda_{2\sigma'}} -1 \right)
\langle \hat{N}_{\text{O}}^{1\sigma} \hat{N}_{\text{O}}^{2\sigma'} \rangle ,
\label{chi_s}\end{aligned}$$ in the relevant energy range $0<E<eV$. The parameters $\lambda$ corresponding to all others terminals are set to zero.

Using Eqs. (\[approx\]), (\[counting\]) and (\[chi\_s\]), at zero temperature, one can calculate the single terminal probability distribution: $$P(Q_{1\sigma})=\left( \begin{array}{c}M\\Q_{1\sigma} \end{array}\right)
\left[ 1- 