Confusion while deriving kinetic-molecular theory of gases 
How many of the molecules will  collide within a given area  A  of the
wall normal to the  x-direction  in  some  fixed  time  interval  Δt ?  On  average  half  the  molecules  have  a   positive x-component  of  velocity.  Therefore  half   the  molecules  contained  inside  a   cylindrical volume of cross sectional area  A  and length
vΔt  will strike the wall within a  given area  A  during the time
interval   Δt.
So, the number of collision with A during $\Delta{t}$ is
$\frac{1}{2}\frac{NAv_x\Delta{t}}{V}$


So my question is how can we assume that half of the molecules are moving one way and half the other? I know that we are focusing on one co-ordinate i.e. x-coordinate, thus, we are not focusing in other directions like up and down etc.Nevertheless,I wondered, if I were the physicist coming up with this derivation, why would I divide the number of collisions by 2 saying that half of the particles go one way and half the other way. How can it be just half not one third any other probability.
Thanks for the time and help.
 A: A simple way of thinking about this is to think about the motion of particles through an imaginary plane somewhere inside the fluid. The number of particles crossing the plane from one side to the other must be the same as the number of particles crossing in the other direction. This must be the case because first of all the gas is homogeneous and it has the same properties (e.g. average kinetic energy, particle velocity, etc.) everywhere. If the particles moved from one side to the other more than in the other direction, you'd have a build-up of particles on one side, violating the second law of thermodynamics.
For particles near a wall, they don't 'know' they are near the wall until they hit it. So the same rule must apply to particles near a wall. 
Viewing it from the perspective of the 2nd law  might be the easiest way of thinking about it, but you can also view it purely from the basic assumptions of an ideal gas: a bunch of non-interacting particles. If you had just a single particle in the box, the same law would apply. On average, over long enough time, the particle would find itself moving toward a wall the same number of times it would find itself moving away from it.
