In an apartment complex with an even number of rooms, half have one occupant, and half have two. How many roommates does the average occupant have?

Question

Here's a little number puzzle question with strange answer:

In an apartment complex, there is an even number of rooms. Half of the rooms have one occupant, and half have two occupants. How many roommates does the average person in the apartment have?

My gut instinct was to say $\frac{1}{2}$, but apparently that is wrong and the correct answer is $\frac{2}{3}$????

I saw this problem on Twitter with very little explanation and wasn't able to find it online anywhere. If anyone could shed some light on this for me that would be awesome : )

Answer 1

If there are $2n$ rooms, then there are $3n$ people. Clearly $2n$ of them have exactly one roommate...

Answer 2

Try a very simple example first: 2 rooms. So, 1 room has 1 person and the other has 2. There are 3 people. 1 of those people has 1 room mate. The other 2 people have 1 roommate. So, 2 out of 3 people have a roommate.

Answer 3

This is a well-known paradox known as the friendship paradox. In this case, your intuition leads you to believe that living in a one-person household is as common as living in a two-person household, because there are equally many of both types of house. But that doesn't translate to their being equally many people of both types.