Website Preview Graphic for Wordpress Site

Question

No idea how to ask this... so I'll give it my best.

I have a Wordpress site, self hosted. When I post a link to a post on Facebook, Facebook tries to detect a preview graphic to show along with the post (see screenshot)

For whatever reason, Facebook always picks up the Stumbleupon share button for the graphic. It's not the first graphic on the page, and it's not the last either, so I'm not sure why.

But more importantly, my question is... when I am not using a Featured Image for the post (sometimes I do, sometimes I do not), how can I make sure that previews such as this pick a different image? The site uses a favicon, but that doesn't seem to work in this instance.

Is there an image, or something I can set in code, that would allow me to specify what image gets picked up when a site tried to create a preview like this?