I want to begin my explanation using abstract mathematical explanation to repetition possibility by taking independent samples $X_n$ from some continuous probability distribution: https://math.stackexchange.com/q/1739927/
If we applied this same principle and its conclusion and assumed that the universe is homogeneous and were able to prove its infinity someway (along with matter inside it including stars and planets) then, statistically, it would mean that the chance of repetition of life again, at least and in the most conservative approach here, as a form of doppelganger extraterrestrial life would be zero? meaning that we are alone in this universe
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No, quite the opposite. If the universe is truly infinite and approximately homogeneous, then I invite you to point in some direction in the sky at night, and if you travel far enough in that direction, you might have to point through several stars and planets and alien organisms to get there, but eventually you might well be pointing at a nearly exact copy of yourself pointing back at you. With infinite repetition things that are possible can become probable and things that are probable can become certain.
Not everything works this way, and it does depend on how “identical” your setups are as you repeat. But the basic math is like this: imagine we model a bridge but we ignore continuous “damage” to it and treat its failure instead as an all-or-nothing thing which has an 0.01% chance of failing in any given day, independent of any other day. One can derive that this bridge will live about 10,000 days or 27 years, plus or minus another 27. It actually in the continuous limit has a 50/50 chance of failing apart after $10000~\ln2$ days or 19 years. So those independent little chances of something happening, under many many repetitions, eventually lead to this very unlikely thing (one in ten thousand) becoming probable at $10000~\ln2$ repetitions and becoming 99.995% likely after a hundred thousand repetitions. Very simple math. And all that the above statement is doing is the same argument, extrapolating an Earth-sized cylinder through space and saying “hey, that this Earth is roughly the way it is (up to your uncertainty) is some unbelievably tiny probability $p$, but let’s chain those cylinders together in the direction you're pointing until we get $10/p$ repetitions and I suppose if you're right that everything is infinite and homogeneous then we'll not run out of universe before $10/p$ repetitions, even for small $p$.”
The “(up to your uncertainty)” is also very important here, because it is what makes the probability finite rather than infinitesimally zero. Mathematicians deal with say real numbers being perfectly specified, so 1.00125267... is very different from 1.00125264... but us physicists have to say “look my experiment has an 0.2% error and I can't resolve the difference between 1 and 1.002, so all of these are approximately 1.000 up to my error bars.”
Actually, something that bugs some people is the idea of “Boltzmann brains.” This is the idea that generating a brain randomly is much easier than generating a whole body and Earth and brain all together, so that if the world really does proceed to a state of maximum entropy, a thermal equilibrium, then there is some sort of recurrence time (in the simplest case a Poincaré recurrence) where the thermal equilibrium randomly generates a tiny bit of brain like yours perceiving the things that you are perceiving but the vast vast majority of times this happens that brain is totally and utterly detached from the real world and vanishes after a moment. And the reason that this application of infinity bugs people is that it means that averaging over all the entities in our universe which are having the conscious experience that we are having of staring at our computer screens on a physics website pondering the universe, almost none of them are actually living on an Earth and almost none of them are going to last longer than the next five seconds and almost none of them are therefore actually seeing anything like the real world, they are just in a very temporary hallucination. And that brings up questions about why we are so sure that we are not.