After paying close attention to the my Flickr badge, I noticed that the selected pictures weren’t as random as I’d like. A quick search on Flickr support forums revealed that I am not alone. Something is wrong with their random number generator.
Gladly, there is a workaround. It’s somewhat painful, but gives far better results. I have changed the badge code to select random images from all my public pictures tagged with “badge“. There are only a few at the moment, but I’ll be adding more soon.