They are presented with situations in which multiple alternatives are possible, and they select one.

This does not cut it. First, you need to additionally specify that they are fully aware of these multiple possibilities. When a man decides to cross the street and gets killed by junk falling from the sky, he didn't choose to die.

Second, the word "select" fully encapsulates the mystery of the word "choose" in this context.

Third (I didn't originally make this clear), I'm not looking for a fully reductive explanation of choice, so the "explaining away" discussion isn't relevant. The statement "Believe me - they choose" appears to be attempting to communicate something, and I believe the payload is hiding in the connotation of "choose" (because the denotation is pretty tautological: people's actions are the result of their choices).

Belatedly I recall prior discussion of connotation. I've edited my original reply to include the flashy LW keyword "ADBOC".