Is there a media/music player program that has a "biased" shuffle?

Try 'Shaken, not stirred'. I had the same question as you did and this little freeware programme offers exactly what I wanted.

In "iTunes DJ" (formerly "Party Shuffle"), click "Settings" and check "Play higher rated songs more often."

(This is not the same as the "Smart Playlists" mentioned in the question... it will still play low-rated/unrated songs.)