a physical question about probability

Question

assume that we have a cubic box which contains a large number of molecules. Therefore we know that the molecules move in different directions and hit the walls of the box . I read somewhere that with a good approximation it is possible to say that 1/3of molecules move in the x direction,1/3to the y and 1/3to z.my question is that what would then be the probability of each walls in X,YandZ direction if we have a cuboid which would be a kind of cube that has been stretched in the x direction for example?

Answer 1

The statement that the molecules are moving randomly in a three-dimensional space is equivalent to saying each of them has $1/3^{rd}$ probability to move along a particular axis (or in other words, that $1/3^{rd}$ of them are moving along each of the three directions). Even if they are in a cuboid the axes remain the same - each of the molecule will still have the same chance of moving along a particular direction (and it doesn't depend on the length of the cuboid you place it in). So still you can make the approximation that $1/3^{rd}$ of them are moving along each coordinate axis. However the probability that a molecule will hit a wall may vary (it will be less for the wall parallel to $yz$-plane in this case).

Suppose we take a cuboid with lengths $1,1,t$ where $t>1$. If probability of hitting the wall parallel to the $xy$-plane is $p$, then probability of hitting the wall parallel to $xz$-plane is $p$, and that of hitting the wall parallel to $yz$-plane is $tp$

$$p+p+tp=1~~~~~\Rightarrow ~~~~~p=\frac{1}{2+p}$$