Well I don't see how anyone can come up with actual probabilities. But can talk about possibilities at least.
First, I think it's probable that any technological ETI is likely to be at least thousands of years, and most likely millions of years in advance of us. That's because, chance of an ETI making a start on technology within the same thousands of years time as us, in the billions of years history of the universe - a tiny one in a million type possibility.
There are other ideas such as ETs last for such a short time that only recent ones would still be around - but seems to me that surely - if they've reached the level of technology where they can travel through space to another solar system - that they are unlikely to be "may fly" like species that die out within a few thousand years.
So, well others might make other assumptions but that's my starting point here.
MEETING ETS "IN PERSON"
Well like most with a background in astronomy, I don't think it is too likely that there are ETs here already zooming around in spacecraft. Reasons are
1. Why haven't amateur astronomers spotted them? Every clear night amateur astronomers are out in their tens of thousands looking at the sky - not just through telescopes, also naked eye observation, and they keep a keener eye on the daytime sky than others also.
So if there were ETs here already in spaceships, then the amateur astronomers would spot them coming and going, just as they spot satellites and rocket launches whenever those happen.
2. For those who think it is a government cover up - okay can see the motivation for the US - just possibly - to cover up. But what is the motivation for the ETs?
Apart from the idea of captured ETs unable to !"phone home" . That I find unbelievable, given the likely technology gap between us and them. With millions of years technology advance over us, their spaceships wouldn't crash. And indeed they wouldn't need to land in spaceships either, they could send invisible nanoprobe spacecraft, any ET spacecraft coming this way would surely have the capability to launch a fleet of spacecraft, of nanometer scale, indeed, probably like mini cells, so small to be beyond the resolution of difffaction limited optical microscopes, coming together in swarms to accomplish tasks. We aren't that far away from that ourselves. May well be less than a century away from it. With a million years more of technology surely that's possible.
So idea of ETs lumbering in on a big clumsy spacecraft and landing on Earth and crashing, and getting captured by us technologically inept humans (from their point of view) - for me anyway - that defies belief.
So - doesn't mean there aren't ETs here. I think it's unlikely. But if they are - then for whatever reason, they have chosen to remain invisible to us, and so - that would depend on whether they change their mind, what might induce them to change their minds about whether to communicate with us.
However, personally I think that's unlikely. Because there is no sign of ETs colonizing the galaxy - well either they haven't - or else - they keep a very low profile everywhere. If they are gardeners say, like the ET in the film, maybe they would do that and you wouldn't see their presence because they travel, explore, but interfere hardly if at all with the galaxy.
So - if there are ETs there - I envisage them as probably like that. Long lived most likely. Mature civilization millions of years old. Not in a hurry. And we might be due a visit some time in the next few thousand years.
If there are none of them close by, it's likely to be a few thousand years anyway, before they get a signal from any probes they might have left in our solar system to alert them to our presence.
If they do visit us in person - then - one major issue would be, compatibility of our microbes with their microbes. Might be that they are intelligent plant-llike creatures - many plants can do fine without anything except their own cells, and nutrients - or for whatever reason don't have microbes that they depend on, and can be sterilized. If not, then they might just send surrogates down to our surface, telerobots.
Or we might meet each other in space, again through telerobots. I imagine by then telepresence would be far ahead of what we can do now, so close to really being there, that I imagine that it might well be the norm, to meet via telepresence, especially with one ET talking to another that needs a different environment or has incompatible biochemistry or hazardous microbes.
And in any case - is obviously much safer. Why do a dangerous mission yourself, when you can send a telerobot? If we do ever meet ETIs "face to face" I expect we'd meet their telerobots - or else semi-automonous robots, rather than meet them in person - at first anyway.
OTHER WAYS WE COULD MEET THEM "ALMOST IN PERSON"
Might be though, that they have left a "von neumann probe" some kind of a machine - with artificial intelligence of some sort.
I'm skeptical that we will make much progress in AI in the next century or so. But given a million years - seems not impossible at all. Something that - maybe is based on understanding, I mean really understanding, how biology works. Something that is half way between biology and machine, maybe has cellular structure, maybe replicates in a way similar to microbes or animals.
If so - maybe there is a "bio-machine" like that, somewhere in our solar system, just lying dormant, which will wake up and start to communicate with us depending on some signal. Or waiting for us to find it, like the monolith in 2001 a Space Odyssey.
So - that does seem possible to me - but totally impossible to assess what the probability is.
OR DETECT THEIR RADIO SIGNALS OR OPTICAL SIGNALS
This then - most astronomers think most likely. But then - sadly - not much chance of two way conversation. Probably take a long time to decipher them once discovered.
In this case - well we have hardly started searching. We could only detect really strong signals, and signals sent our way intentionally, at any distance. And ET civilizations are likely to evolve to be more and more efficient at use of radio waves and optical communication.
So - could happen, probably most likely if some ET is intentionally broadcasting a signal out to the galaxy to try to attract the attention of young emerging ETs like ourselves.
If so - hopefully the message they sent is carefully thought out to be useful to us. That's the general idea of Carl Sagan's "Contact" basically, though with a lot of other things involved as well.
OR UNINTENTIONAL SIGNALS
Or, we might detect ETs but not through an intentional signal.
One way is through search for mega-structures, as in the search for Dyson Spheres.
If so - well that might pay off in the next 100 years if they are there.
If so - well we might see a strange infra red signature, and dawning realization that it has to be a mega structure. But we'd know next to nothing about the ETs that constructed it unless they sent messages that we could decipher.
So that would be fascinating, but also frustrating.
If we also get messages from them, its possible that it might be really hard for us to understand.
That's always a possibility. Someone has to be first even in the entire universe. So - we might be first in our galaxy. Or might be first in our universe.
OR FIRST WITH TECHNOLOGY
Or - might be that we are the first with space technology. For instance could be dozens of ETs that can't build spaceships or radio - for every one that can. Just because they are, for instance, not strong enough, not dextrous enough (like parrots) - or live in water permanently (like octopuses) - or they live in underground seas like the Europa seas and don't even know, perhaps, that there is any more to the universe than their home ocean. They could be millions, or billions of years old civilizations and still not have spacecraft.
In that case, we might meet ETs in surprising ways.
One possibility, rather fun, is that there might be ETs living in the ocean of Europa (moon of Jupiter) in our own solar system - even a millions of years old civilization. We'd never know, not yet, not if they don't have radio or such like, and they wouldn't know about us either if they didn't have technology to "see"through the ice that covers their ocean.
If so our first encounter of them might be when we send a probe into the Europa ocean and find ETI fishes or octopuses or - manta ray type creatures, or seahorses or frond like creatures, or whatever they are clustering around it in curiosity :).
Hard to judge the probability of any of these ideas, as we haven't got anything to go on.