Probability of picking multiple matching pairs from a weighted distribution.

[Gabeybaby]

Hi, I'm stumped and cannot find any articles or discussions about this. The closest I can get is the birthday paradox, but most people give each day an equal probability in those examples.

-
Let's say we have 200 names weighted according to their popularity. (names like John and Mary have a higher weight)
For simplicity, let's say those are the only names that exist.

If we gather 50 people in a room, what are the chances that there are 5 pairs of matching names? (5 pairs -- 10 people)
-

How might I set this up?

 

[Romsek]

Fundamentally it's pretty simple.

List out all possible distinct arrangements of 50 names from an alphabet of 200 with (at least? exactly?) 5 pairs of matching names.

For each arrangement use the weighting scheme to determine the probability of that arrangement.

Add all these.

If you are after at least 5 pairs of matching names it's probably easier to find the probability of 4 pairs of matching names or less
and subtract this from 1.