I was reading a derivation of the rate equation from collision theory and there is one thing which confuses me a bit. In the derivation we assume that a particle will collide with every particle within its "collision volume" but it seems to me that this is a gross overestimate of the actual number of collisions. My reasoning is that if the particle successfully reacts during the first collision, then it will have only made 1 collision, and similarly with successive collisions. So shouldn't we instead be using an expected number of collisions rather than just saying the particle will collides with everything it has access to. This problem is then emphasized because we next multiple the number of collisions by the number of particles which seems to just blow up the error further. Am I getting confused on something here?