Here is a great article I found a few months back on waitbutwhy.
Here’s a few juicy quotes to get you started (but really, just read it).
As many stars as there are in our galaxy (100 – 400 billion), there are roughly an equal number of galaxies in the observable universe—so for every star in the colossal Milky Way, there’s a whole galaxy out there. All together, that comes out to the typically quoted range of between 1022 and 1024 total stars, which means that for every grain of sand on Earth, there are 10,000 stars out there.
Essentially, the Fermi Paradox is looking into the probability of earth like planets, and the subsequent probability of sentient life. The conversation gets really, really interesting from there.
Moving forward, we have no choice but to get completely speculative. Let’s imagine that after billions of years in existence, 1% of Earth-like planets develop life (if that’s true, every grain of sand would represent one planet with life on it). And imagine that on 1% of those planets, the life advances to an intelligent level like it did here on Earth. That would mean there were 10 quadrillion, or 10 million billion intelligent civilizations in the observable universe.
Based on our own solar system history, which is relatively young, one would expect there to be civilizations far more advanced than ours. Because we haven’t detected any such sign of intelligent life this could signify something strange going on. He goes on to explain the difference between a type I, II and III civilization. If a type III civilization where to exist, which we would assume should, then we would know about it, because a type III civilization would harness the energy of the entire galaxy.
One hypothesis as to how galactic colonization could happen is by creating machinery that can travel to other planets, spend 500 years or so self-replicating using the raw materials on their new planet, and then send two replicas off to do the same thing. Even without traveling anywhere near the speed of light, this process would colonize the whole galaxy in 3.75 million years, a relative blink of an eye when talking in the scale of billions of years:
Continuing to speculate, if 1% of intelligent life survives long enough to become a potentially galaxy-colonizing Type III Civilization, our calculations above suggest that there should be at least 1,000 Type III Civilizations in our galaxy alone—and given the power of such a civilization, their presence would likely be pretty noticeable. And yet, we see nothing, hear nothing, and we’re visited by no one.
Welcome to the Fermi Paradox. Read on to find out why we might be alone in the universe or how we might be first to become a type III civilization.