From my lecture notes (ICL, dept. of Physics) it is written:
Consider a thin piece of material with thickness $d$ containing target particles with number density $n$, as illustrated in Figure $\bf{2.3}$. Each target particle presents an area, the cross-section $\sigma$, for the reaction. For a front surface area $A$, there are $Adn$ targets, which therefore have a total target area $Adn \sigma$. Hence, the probability of an incoming particle hitting one of the targets is $Adn \sigma/A = dn\sigma$. Therefore, in this ‘thin target approximation’ the reaction rate per unit area of target is simply: $$R=\phi d n \sigma$$ where $\phi$ is the incident particle flux (particles per unit area per unit time); the point is that $d$ and $n$ can be different for different experiments, but $\sigma$ is the physical property we compare. Note that this only works for thin materials, meaning $d$ is small enough that the probability of a reaction $dn \sigma \ll 1$. Otherwise, every incoming particle will see several targets and have multiple interactions. In this situation, a more appropriate measure is the mean free path, $\ell$, of the particle in the material. Clearly, on average the particle will move this distance before interaction ($\color{red}{\text{corresponding to a probability of one}}$), so the mean free path is given by $\color{red}{\ell n \sigma=1}$, or $$\ell=\frac{1}{n \sigma}\tag{1}$$
I don't understand the parts in red. How does it correspond to a probability of one? I looked on page 34 of W.S.C Williams "Nuclear and Particle Physics. Published by Oxford science publications 1991. But the formula, $(1)$, is simply stated without proof. But Williams does mention the kinetic theory of gases, so I searched the internet and found this which looks nothing like $(1)$. The closest to a success I have so far is Wikipedia (which even uses the same setup and diagram as ICL). But still, there is no clear derivation that $\ell=\frac{1}{n \sigma}$. Does anyone know of (rigorous) proof of $(1)$?