Quantum Mechanics

I haven't studied physics in years, and when I had studied physics, I only took a few introductory classes in it. I am hoping someone more knowledgeable will be able to answer my questions.

In quantum mechanics, electrons aren't in a specific location around an atom's nucleus. Instead, there is a probably density function that provides the location. There are places that the electron is likely to be, namely close to the nucleus, but there are places that the electron could be, like very far from the nucleus, but is actually very unlikely to be. Given the overwhelming number of electrons that exist, an extremely improbable event like the electron being very far from the nucleus is likely to occur occasionally.
How often do these events occur? What is the effect of it when it occurs?

The great Douglas Adams used these concepts in the Hitchhiker's Guide to the Galaxy books. I was hoping the popularity of his books would spur some research into the topic, but I haven't found anything useful yet.