Is Eternal Self-Awareness Inevitable in an Infinite Universe?

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For an entity to be born and recognize itself as a self-aware entity ("I exist"), there had to be a non-zero probability that such event occurs. In a universe which is known to host some 100 billions of galaxies - and which possibly is infinite or even just one universe among an infinite number of universes - wouldn't that make sense then to say that the probability of such event occurring again is inevitable?
Obviously, there won't be any relation between the 'I' in the present and the next 'I' in which self-awareness occurs (doesn't have to be human), and no continuity of memory. That means that the new entity won't have a perception of any time lapse between the 2 events, whether it be 5 mn or 1 million Earth years. In that sense, can we say that any self-aware entity like me and you will inevitably experience self-awareness for as long as the universe sustain life?
I wonder if my reasoning on probability makes sense, or whether I fall into some kind of flying teapot fallacy... I'm interested in reading your thoughts on this.
 
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wouldn't that make sense then to say that the probability of such event occurring again is inevitable?
The probability could be tiny, and even decreasing with increasing age of the universe. In a finite universe, the expected number of future self-aware entities can be very small (<<1).
In that sense, can we say that any self-aware entity like me and you will inevitably experience self-awareness for as long as the universe sustain life?
I don't think so, as there is no relation between us and some self-awareness far away in time or space.
 
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