At the beginning of Fallout the Vault 13 Overseer sends the player to replace the vaults water chip, and points out Vault 15 might have a replacement. After the player defeats the Super Mutant threat and reaches the end of the game, the overseer exiles you out into the Wasteland out of fear that you might influence the other inhabitants of the vault, similar to the predicament in the Trouble on the Homefront quest in Fallout 3.
Despite this many people from the vault followed the Vault Dweller and established the town of Arroyo. In this game you are the Chosen One, a direct descendant (grandson/great grandsom) of the Vault Dweller. You are sent to Vault 13 once again to find the Garden of Eden Creation Kit or GECK to save your town.
Also, in Fallout 2, the Enclave captures the inhabitants of Vault 13, and because they had been cut off from the rest of the world for 200 years, it made them ideal test subjects for their experiments with the F.E.V.
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