# The number of mean free paths which a particle traverses is independent of the material traversed

This is a manual from a Monte Carlo physics manual:

Computing the occurrence of a process

Can anyone tell me why the number of mean free paths of a particle is independent of the material which it traverses? It does not make sense to me.

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It is the same as in radioactive decay: The amount left after three half-lives passed is always $\frac{1}{8}$ or the original amount, no matter which radioactive material you look at. But the actual time that passed is different for each material i.e. $3\tau_{\frac{1}{2}}$ where $\tau_{\frac{1}{2}}$ contains all the material dependent information.
$n_\lambda = \int \frac{dx}{\lambda(x)}$
both $dx$, the physical distance travelled, and $\lambda(x)$, the mean free path, depend in the same way on the material, so the end-result does not.