Linsey,
You're not actually getting a random child of a particular page here, of course. You're getting a random instance of a particular class. That's fine if all instances share the same parent.
I've just used this code to generate a random testimonial on a home page: pretty much the same use case as you have, it appears.
I can't offer code to produce a random child right now, as I'm too trying to get a site ready. Just wanted to warn people.
Toby