Given a number of four-person families, how many people need to be selected so 50% of the time, half of them are in familes with another selected? Doing some worldbuilding and was thinking about how connected my world is. I had thoughts on dealing with how I could calculate pedigree collapse and such, but wanted to use a simpler model to think about for now, so:
Given a number of four-person families, how many people need to be selected so 50% of the time, half of them are in familes with another selected?
I know that it's obviously at least double the number of families, a.k.a half of the number of people, and less than everyone except two.
For 1 family, that's the trivial case. Any two people will do, and that's the minimum number for the selected to be in families with each other.

Suppose people are selected independently with probability $p$.  The number of people $M$ selected in a family of four has a $M\sim Binomial(4,p)$ distribution, and you want $P(M>1)/P(M>0)=1/2$
In code
curve(pbinom(1,4,x,lower.tail=FALSE)/pbinom(0,4,x,lower.tail=FALSE),
  ylim=c(0,1),ylab="P(another selected in family)",
  xlab="proportion selected")
abline(h=1/2,lty=2,col="purple")

If 1/3 of the people are selected, your probability is just over 1/2.  I might suspect there's a more elegant reason why it's 1/3, except that it isn't exactly 1/3. Numerical optimisation gives 0.32809

