Beatrice decides to jump first in the book, Divergent, because she is very aware that no one there things she has chosen the Dauntless faction well. In fact, having come from Abnegation, they don't expect her to have courage, ambition, or any other strong emotion. She shows them all by volunteering to do something that completely terrifies her (and the rest of the initiates). She jumps first because she feels that she has to prove something to them.
