A simple model for the beginning of the universe: $n$ particles are created. Every particle can be either matter or antimatter, with probability $\frac{1}{2}$. After the particles are created, matter and antimatter "cancel each other out" until particles of only one type remains, in a 1:1 ratio. The question: For a given amount of particles $k$, what is the probability that this amount will remain?
If X is the number of matter particles, we have that $K=|2X-n|$, so call $Y=2X-n$. Since X is binomially distributed, $E[X]=n/2$ and $V[X]=n/4$ and so $E[Y]=0$ (as expected...) and $V[Y]=2^2V[X]=n$. Using Chebyshev we get $P(K>k)\le\frac{n}{k^2}$.
This seems to imply that the amount of particles in the beginning of the universe (in our simple model) should have been at least quadratic of the amount of particles existing today. However, I feel this result is weak and can still be improved, maybe using a variant of Chernoff I'm not thinking about? My "gut feeling" is that as $n$ grows larger, the system behaves in a less chaotic way and the probability of "almost zero" values of K increases, maybe even to 1. Am I correct in my guess, and if so - how to prove it?