No life jacket?
You don’t wear life jackets on a cruise ship. Life jackets are stored in the life rafts, and would certainly be lost or unavailable if issued to passengers with the understanding that they wear them 24/7. There is a man-overboard device that has a light, radio beacon, and radar reflectors, and is deployed (thrown overboard) whenever there is credible report of a passenger going overboard. If you can swim to the device it functions as a life raft, if not, it guides rescuers to the last know location of the victim.
No one wears a life jacket on cruise ships and there wasn’t time to grab one as he fell overboard.