Picture a box with no partition in it, and draw a line down the middle, nothing more. Drop one particle in and whether it is in the left half or the right half is not decided at all. Both happen equally often.
Two particles are no better. There are 2^2 = 4 ways for them to fall left or right, and both being on the left is one of those four. One time in 4 is hardly a rarity. With three it is one way out of 8, which still turns up perfectly often.
So far we cannot say anything about what is inside the box. Yet as the number of particles grows, the picture changes completely. All we change is the number. Not one new law is added.
There are \binom{N}{n} ways for n of the N particles to be on the left. If we count one arrangement, with every particle's side settled, as one case, then every case happens as often as every other, so the n with the most cases is the most likely n.
The figure lays those counts out in order of n. At N = 4 the bars are nearly level and every n happens often enough. As N grows the middle rises and the ends lose any height you can see.
Everything on the left, n = N, stays one single case however large N becomes. The middle, n = \dfrac{N}{2}, grows by orders of magnitude along with N. The ends do not grow and the middle does, so the two stop being comparable.
How far the peak spreads is measured by the standard deviation of the departure from the centre. The spread in the number on the left is \dfrac{\sqrt{N}}{2}. It grows not as N but only as the square root of it.
Multiply N by 100 and the centre of the peak, \dfrac{N}{2}, moves out to 100 times its place. The width, meanwhile, grows only by a factor of 10. The width cannot begin to keep up with the centre, so the peak narrows on the stretching axis.
The figure draws the peak while stretching the horizontal axis along with N. The width really is growing, and yet against the stretching whole it thins to a needle.
What matters is not the width itself but the width as a fraction of the whole. Divide the width \dfrac{\sqrt{N}}{2} by the total N and you get \dfrac{1}{2\sqrt{N}}. That is the measure of how far from an even split things can look.
At N = 100 it is 5\%, at N = 10^{6} it is 0.05\%, and at N = 10^{23} it is around 10^{-12}. No instrument can measure an imbalance of 10^{-12}.
The figure shows how that fraction falls. The fall of \dfrac{1}{2\sqrt{N}} is not itself fast. It ends up immeasurably small despite not being fast, because the N we are dealing with is larger by so many orders of magnitude.
With 10^{23} particles the two sides never depart from an even split, for all practical purposes. Not that it cannot happen: the fraction of the time it happens is simply too small to measure. Which also means we may use a probability as a flat statement of fact.
Once that is settled there is no need to follow the particles one at a time. Choose the single most likely state and look only at the narrow band around it. That is why we are spared solving 10^{23} equations of motion.
This is where statistical mechanics starts. What remains is how to count that likelihood, and that is where we go next.