According to Ted Demme, Vincent Pastore arrived at the audition in character dressed as he is in the film and threatened to harm Demme if he did not get the part. After a scared Demme gave him the part, Pastore broke character and politely thanked him.
According to Ted Demme, he did not have the budget to film an actual horse race. That is why the viewer only sees Harry's face as he is watching the horse race.