Coincidences are observed routinely, no belief required.

Coincidence is simply hindsight, or, oh, wait, just a coincidence.
What is there to "believe" about it?

Wow! I saw this post just as I was thinking of posting the exact same question!

In 11th grade, I took a trip to Chicago during spring break. On a trip to the Museum of Science and Industry, I'm walking with my Dad on the 2nd floor when I spot a teen girl with red hair walking the opposite direction. She stares back at me; I stare because she looks just like a girl that I share a bunch of classes with. Sure enough, it was her, both of us 2000 miles from home, but passing each other. How often does that happen but we don't even know?

As nouns the difference between coincidence and serendipity is that coincidence is while serendipity is an unsought, unintended, and/or unexpected, but fortunate, discovery and/or learning experience that happens by accident.

An example of coincidence would be an unplanned meeting with a friend when both of you happen to be on the same street and at the same time walking towards each other on any given day.

Serendipity usually refers to finding something that you had not been looking for but is nonetheless usually a pleasant surprise.

What else is there?

It's called "serendipity".