Alright, I'll toss in a few assumptions.

So, I suppose that we have two teens, picked one in the US and one in Ireland, that get randomly matched as pen pals. They talk and talk, until one of them says that they're adopted. Hey, the other one is also adopted, this must be an incredible coincidence! But is it?

Let's do the math. So, the row for 2003 is actually missing in the table you linked, but we're just giving an estimate, so let's say it's roughly the same for the neighbouring years, and assume 36 adoptions per 1000 births. This means that roughly the $3.6\%$ of kids are adopted. We don't have the data for Ireland, but let's keep the same rate, for simplicity.

What are the chances that both kids are adopted? Well, that's simply $0.036 \cdot 0.036 = 0.001296$, which gives us an estimate of about $0.1 \%$ (the adoption rate is an estimate, so we can't get a precise answer). Quite rare, right?

The fallacy here is, the question only came up because one of them brought out the topic. Which means, if neither of them was adopted, they wouldn't have realized this. So, a better question is: what are the chances that both kids are adopted, if we know that at least one of them is adopted? Well, the chances that at least one of them is adopted are \[ 0.036 \cdot 0.964 + 0.964 \cdot 0.036 + 0.036 \cdot 0.036 = 0.070704, \] so the chances that they are both adopted, knowing that at least one of them is adopted, are \[ \frac{0.036}{0.070704} = 0.0183299389, \] which gives an estimate of about $2 \%$. Still rare, but not as much as before.

But hey, that's just one of the many possible "incredible coincidences" that could have happened! Wouldn't it be an incredible coincidence if they both have a twin sibling? Or if they are both named Alex? Or if they both went to a school named after Nikola Tesla? Or if they are both born on April 4th?

The point is, there are so many possible "incredible coincidences" that one of them is pretty much bound to happen quite often. This means that an even better question would be: what are the chances that we both have some rare thing in common? While giving a precise answer is impossible here, it's safe to say that it's quite high.

See the birthday paradox for reference: the chances of having two students born on the same day (what an incredible coincidence!) in a class of 20 students are above 50% (https://en.wikipedia.org/wiki/Birthday_problem), and this question is not unlike that. Having a specific, rare thing in common is extremely unlikely, but having some rare thing in common is actually quite likely.

I hope the answer is satisfying!

This question is very vague. It is difficult to provide a mathematically sound answer to this question.

One would need tons of assumptions to give a numerical answer. After some point, too many assumptions in a model are just self-serving.

Okay, apologies, this is my first post. How would I make it more specific? In this situation, both of the teens were adopted and surprised to find this out which is why one of them posed what are the odds of them meeting. Imagine it were a conversation you were having with a new acquaintance, would it be possible to work the math out? How would you go about doing it?

Are you asking for the probability of two wins being pen-pals given that they were separately adopted, or the probability that two orphan twins get adopted and also end up becoming pen pals?

Separately adopted, one in the US the other in Ireland. They were randomly paired/met as pen pals and had no prior relationship.

I think I can give a reasonable answer to this question, but I have to dig for data and write quite a lot of stuff, to justify the assumptions and get rid of some bias. It's too much work for a 10$ question. For the current amount, I can maybe provide a sketch.

How much is still an assumption? Please forgive me. I am not trying to be obtuse I am willing to talk this out so there is less assumption and more accurate data. Also, what would a more reasonable bounty be?

Well, it depends on the data I can find, but anyway the problem is that the question is not well-posed imo. Assuming the spirit is "I met this person and hey, we're both adopted, what a coincidence!" I can try to do the math, explain why it's not that strange, and go through a few related paradoxes. Regarding the bounty, I'd say somewhere between 20$ and 40$, depending on the level of detail that you want.

If you can provide data about the adoptions/births ratio in the two countries in 2003, that would result in a more accurate answer, and save me some time I can spend to get more into detail instead.

I'm working on the bounty. US info was super easy https://www.johnstonsarchive.net/policy/adoptionstats.html Ireland is a bit muddier.

Bounty increased. Thank you for the input.