How did the Big Bang cause the cosmos to be infinitely large? If the cosmos is product of the big-bang as we think then it would seem there is a problem with the probability of it's supposed infinity (as it was initially finite). In addition, the rate of it's expansion is also finite. On account of this, how can we consider it to be infinite?
 A: The biggest hole in your argument is the following: "by the theory at first it was finite." This is perhaps one of the biggest misconceptions about big bang cosmology that exists. The big bang theory, as it stands today, merely notes that, at some point in cosmic history, the universe was very dense. This does not imply finite-ness nor does it imply that the universe started "at a point," as so many like to say. The standard model of cosmology simply states the following:


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*Assume I have two objects that are both at rest and are a distance $d$ away from each other at some time $t_0$. Then at a time $t$ they will be a distance $a(t)d$ away from each other, with $a(t_0)=1$ by definition.


By means of general relativity and a few assumptions, we can find a differential equation for $a(t)$ given the type of matter we see in the universe (see https://en.wikipedia.org/wiki/Friedmann_equations). A generic feature of these equations is that, at some point in the finite past (call it $t=0$ -- we might as well), we must have $a=0$. That is, every bit of matter was infinitely close together.
This, of course, is absurd. The problem is that people take this statement very literally. Any self-respecting physicist will tell you, of course, that the point $t=0$ (where $a=0$) cannot be described by our current models of physics. At extremely early times, it's possible that the very concepts of space and time break down. We simply have no idea what happened before a certain time. After a long enough time (around $10^{-32}$ seconds, give or take), the universe is cooled down enough to be described by standard physics.
Note that if we have an infinite universe today, the only point in time at which the universe is finite is exactly $t=0$. Immediately after, the universe is just as infinite as it ever was. The existence of a singularity at $t=0$ does not signal that the unvierse was finite at some time in the past. It simply signals the fact that the universe was very dense at some point in the past. It expanded at every point and cooled and here we are today.
I think this video does a decent job of putting what I'm saying into a visual perspective: https://www.youtube.com/watch?v=q3MWRvLndzs
I hope this helped!
