Respectfully, yes, I do believe it happened by chance. 3.5 billion years, combined the fact that radiation causes strange things to happen.

http://www.medicinenet.com/script/main/art.asp?articlekey=47820

Have you considered all the types of genetic disorders? These are random and can be caused by physical exposure. For example, smoking while pregnant, diseases from mosquitos, drinking while pregnant. So genes can be changed by physical substances.

https://en.wikipedia.org/wiki/List_of_genetic_disorders

So genes do get modified in abnormal ways. And in addition to that, human now are determining how to modify them. Look at these glow-in-the-dark cats.

https://blogs.scientificamerican.com/observations/jellyfish-genes-make-glow-in-the-dark-cats/

So, here's my point.

Radiation causes strange things to happen in cells. Physical chemicals cause genes to change. These changes can cause things like glow-in-the-dark cats.

Now, if suddenly glow-in-the-dark cats survived better than not-glow-in-the-dark cats for some reason, then we would expect to only find glow-in-the-dark cats. Only the strong survive. This is called natural selection.