I don't think this is the classic birthday problem, but if it is, I'd love it restated in a way that makes it sound more like the others.
Basically, I think it breaks down to the probability that two given sets of people (of size 5 in my case) have $n$ common birthdays between them. Members of my wife's and my nuclear families (when we were young) happen to share three birthdays.
When queried as "What is the probability that I would meet someone who's family and mine share three birthdays?" (that is in a personal "my" context, with predefined beginning birthdays to match), it would seem to reduce it to a simpler probability (1 in 365 x 365 x 365
or 1 in ~48M
?), though I imagine it would depend on the size of the sets, and thus be a permutation of some sort(?).
I'm guessing you start with a set of 5 dates (for each of my family members), and then take a total population (perhaps of the US, where I live), but how would one proceed from there to find another family with $n$ common birthdays? You might bring statistics in to make sure you're only looking at other families with at least $n$ members, but I can't shake the feeling that since I don't see this asked anywhere, it's either incredibly simple or still reduces to the classic birthday problem. What is this probability?
TL;DR:
I'll try to state my query as clearly as possible:
What is the probability that I, as a person in a nuclear family of size $m$ will meet someone in a family (of any size $>n$) that has $n$ common birthday dates with my family?
We can leave the probability of falling in love and staying together with this person to a separate discussion.
Thanks for your consideration!
Note: this is what I thought I was submitting the first time, but somehow a previous cached version is what got submitted - apologies for the badly formatted and incomplete question