TL;DR is it would probably not sink after a forward starboard collision with an iceberg below the waterline alone. If a capsize occured, which would be highly likely, it would most definitely sink.
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With rigorous safety standards across the board, cruise ships are some of the safest places to spend your hard-earned holiday. The chances of your cruise ship capsizing or sinking are infinitesimally rare. According to the New York Times, only 16 ships have sunk since 1980.
Over the past 100 years since the RMS Titanic sank in 1912, only 18 cruise ships and some ocean liners have been publicly known to have sunk. And, over the past 50 years, only four cruise ships have sunk while navigating on a cruise.
In such cases, cruise lines have two options: head to a different port up or down the coast to disembark passengers or remain at sea until the ship's home port reopens.