Rotate through videos in single element?

I have a video placed on a page using the Video media element. What I’d like to do is display a single video just as the element currently does. However, when the page is reloaded, I’d like a different video to show up in that spot. I’d like to cycle through a set of videos, but only ever display one at a time. I don’t want to use a slider.

Is this possible with the video element?