Regarding snowflakes, it turns out someone has done the math. You can see it here, along with the underlying reasoning:

Is it really true that no two snowflakes are alike?

His conclusion is: And thus it's unlikely that any two complex snow crystals, out of all those made over the entire history of the planet, have ever looked completely alike.

It's a fascinating website. Here's the main page: http://www.its.caltech.edu/~atomic/snowcrystals/. You can also get there by www.SnowCrystals.com.

When you get there, click on one of the photo galleries, then click on a snowflake image to enlarge it.