However big the universe gets, it's not infinite. Figuratively, it's very big indeed.
We've got 4 places within the solar system which can easily support microbial life.
We have discovered many ways in which
self-sustaining RNA arise naturally. That's not a "one off" phenomenon,
it's just simple chemistry. But one factor is special, the continuous survival of these RNA to form more complex structures for a long period of time. That might not be very common, but given the vast variety of ecospheres that exist in the universe, it's very likely that there is one or more places which are more nurturing for these RNA. So, microbial life may be very common, indeed.
When you talk about intelligent life, things get complicated. Firstly, super-human intelligence is not favoured by natural selection. We just have to look around ourselves, an animal as complex as a chicken is doing sufficiently well with the level of intelligence it has. Still, there could be lots of cases where an intelligent civilization does come about, but it doesn't live long enough to attain radio technology. There have been 5 mass extinctions in the history of our planet, in one event 96% of life on the planet perished. There could be places where intelligent civilizations can survive long enough to attain technology, as soon as they do that, it's very probable that the civilization could self-destruct by technological means long before they become a space fairing civilization. The human species has only been around for 200,000 years, and that's a blink of an eye even on geological time scales, there has been at least one case where we've come close to wiping ourselves out. Furthermore, we've only been technologically capable for only a few centuries. That's merely seconds on cosmological time scales, the universe is 13.7
billion years old.
So, conceptually that gives a us a very limited window of time in which a possible intelligent civilization can exist. It's one way to look at Fermi's question "where are they?". I think, life is common in the universe, even some "intelligent life" like small critters maybe. But I am pretty sure that microbial life is very common in our galaxy not just all of the universe. When we talk about searching for life, we can only meaning fully do it within our galaxy for now. All of this in JUST ONE GALAXY. There are probably billions of galaxies out there, currently we don't have the capability to search for life on those galaxies. Our galaxy is pretty big, and it has tons of stars. We only found the first planet outside of the solar system back in the 90s.
Now, we're finding them in greater numbers than ever before. So, we're still searching our own house, and there's lots of places to look at before we venture out.
Topic: Infinite universe and alien life