What is the linear attenuation coefficient and how does it relate to interaction probability? I have  misunderstanding the linear attenuation coefficient (L.A.C) concept. As known, L.A.C  is depend on absorbed medium and energy of incident radiation.  Supposing,  L.A.C= 100 cm-1, how can this parameter measure the probability of interaction per unit of length however the probability values are  between 0 and 1.
 A: For a LAC $\mu$, the probability of interaction after a path length $l << 1/\mu$, is approximately: $P \approx l \cdot \mu$, e.g. if $l = 0.001 cm$ and $\mu = 100 \, \mathrm{cm}^{-1}$, then the probability of interactions is approximately: $P \approx 10\%$.  This no longer applies when $l \gtrsim 1/\mu$.
More precisely, the probability of interaction (or the fraction of incident radiation which will interact) is:
$$P = 1 - e^{-l \cdot \mu}$$
Or equivalently the Transmittance is,
$$T = e^{-l \cdot \mu}$$
This is often expressed using the mean-free-path $\lambda = 1/\mu$, such that typically a photon will interact after a distance $\lambda$.
A: The linear attenuation coefficient $\mu$ is defined using fractional beam absorption, i.e.
$\mu(x) \equiv -\frac{1}{I} \frac{dI}{dx} $
where $I$ is the intensity of the beam. Rearranging a bit leaves you with:
$\frac{dI}{I} = -\mu(x)dx$
The right hand side can be interpreted as the probability of interaction per unit length. It is properly constrained between 0 and 1 because no change in intensity can be greater than the total intensity $I$. Integrating both sides will give the probability of absorption over some distance $l$ which DilithiumMatrix covers nicely in their answer.
