With the recent redesign of Youtube they've taken out the randomize option, so you can set your playlist to autoplay, but it will always play in the same order. Boring, isn't it?
Luckily, there is a handy site which allows you to play/shuffle a Youtube playlist in randomized order. It does not have a fancy design, but it does the shuffling well.
You just give your Youtube user name, it lists your playlists and you can choose one to play in random order. It even have a Play Next button if you want to skip a number.