How Do You Calculate Collision Probability in a Particle-Filled Volume?

[Zaphodx57x]

In a volume V there exist N particles of diameter d.

If we place some detector in this volume, what is the probability that the detector will measure a collision in a time interval t. Or what is the probability that the detector will travel some specified distance into this volume.

There is a lot of room to make estimations and assumptions in this problem if needed. I'm trying to figure out the general method to attacking these "sorts" of problems, not solving the one I stated explicity.

This is the root question of several problems I've ran into in the beginning of a couple of astrophysics books. I just don't know how to generate a probability out of this information.

Its easy to find the mean free path from the number density of particles and the cross-section of the particle. But how do you generate a probability out of it.

 

[Astronuc]

What is the volume or cross-sectional area of the detector, and are the particles uniformly distributed?

One method is to assume a cross-sectional area of the detector and determine the number of particles passing through that area per unit time. One employs some "cross-section" of interaction.