The One-Third Fuse Logic Problem

Problem: Does Randomly Burning One-Third (1/3) of a Fuse Approach 20 Minutes?

You have a fuse that burns for exactly 60 minutes from one end to the other.
The burn rate is uneven along its length, some parts burn slowly, others fast.

You do the following experiment:
You cut that single fuse into N equal pieces.

Your intuition says:

> As N becomes large (cutting into many tiny pieces), burning 1/3 parts of them gets closer and closer to 20 minutes, because picking one-third of many tiny pieces should “average out”.

 Your tasks:

Please mathematically prove your intuition is right or wrong 
(You don't have to consider the physical world limit - more specifically even though you have so many N pieces, you can burn them 1/3 parts in a row without any time delay)