Open your playlist on YouTube and copy its link from the address
bar (on the YouTube app: Share, then Copy link).
Paste the link in the box below.
A blue box will show your personal Shufflemark link. Bookmark it
(press Ctrl+D, or ⌘+D on a Mac; on a phone, use the browser's
Share or menu button and choose Add Bookmark).
Whenever you open that bookmark, a random video from your
playlist starts playing.