Why did Prospero allow the marriage?

Prospero's decision to let Miranda and Ferdinand marry was made even before Ferdinand came to the island, and was made because the marriage would secure Prospero's position back home, and would make his daughter queen as well. The work Prospero made Ferdinand do, coupled with the enchantment that he put his daughter and Ferdinand under so they would fall in love, merely assured that Prospero's plan would succeed, as it finally does.