 Solution

For statistically independent Thomson scattering events, the probability of no scattering over a proper distance $s$ is $\exp (-n_e{\sigma }_{e}s)$. The photon mean free path in an ionized universe is thereforeThis is about half an astronomical unit. It is a proper distance at that epoch, not a present-day comoving distance. The Thomson scattering approximation is appropriate because the electrons are nonrelativistic and typical photon energies are much smaller than $m_e{c}^2$. The source's spelling of the process is a typographical error; the physical process is Thomson scattering.