Brian Cox shares the most popular theories of why haven’t we found aliens? Sharing because it allows me to expand some salient points that are not mentioned, but need to be stated for clarity.
Assumptions
1. The speed of light sets the maximum rate any species can expand, and this, along with the inverse square law, would limit detection too.
2. The average distance between stars is approximately 6 light years.
3. Space is very big. The volume of the milky way is approximately 785,398,163,397,448 light-years. That's 785 trillion light years.
So if you divide the average distance between stars into the volume, then we get 130 trillion years of travel time. Now that sets the maximum time it would take to visit every star, and of course we, or a hypothetical alien species, wouldn't need to visit every star.
I've discussed this on my other blog where I bloviated on Aliens in BattleTech, and four years ago on this blog about the Fermi Paradox. If you click the latter, I gave a ball park figure of 31 billion years to colonize the galaxy at slower than light speed.
Obviously, the assumptions will generate a different range of guesses, but that's the best we can do.