This feature can be enabled for a playlist. A random video from the playlist will autoplay at a random playback position when you open the page with the playlist.
You can also optionally add the “Refresh” button to let user switch to another random video at a random playback position.
To use the feature first enable it in Post Interface Options in wp-admin -> FV Player -> Settings.
Then the checkboxes become available in FV Player editor:
Here’s a demo of the feature: