Suppose there is a city with population $n$. Suppose also that, for each person, there's a chance $p$ that their soulmate is living in the same city.
In this city there's a club with $m$ members. What's the chance that there are no pairs of soulmates in this club?
Here's my answer
Let's analyse a person who belongs to the club. They have a chance $1-p$ of having their soulmate in another city. If their soulmate lives in the city, then there's a chance $\frac{n-m}{n}$ of the soulmate not belonging to the club.
There are $m$ members in the club so the total chance of not having any pairs of soulmates is:
$\left( 1-p + p\frac{n-m}{n}\right)^m=\left( 1 +p\frac{n-m-n}{n}\right)^m =\left( 1 -p\frac{m}{n}\right)^m$
Here's my doubt
I assumed all events are independent, but clearly if $m-1$ persons don't have a soulmate in the club, then the last person can't have a soulmate in the club either. If we assume soulmate is a symmetric property, then what's the proper solution?