Six Lanterns on a Carousel

Six lanterns hang evenly spaced around a carousel. Exactly three consecutive lanterns are lit and the other three are dark, with all $6$ rotations of the pattern equally likely. You may walk up and check lanterns one at a time, in any order, adapting to what you see. Afterward you must predict the state of every lantern you did not check, earning $\$1$ per correct prediction (checked lanterns pay nothing). How many lanterns should you check, and what do you expect to earn?