Quant Interview Question #quant

Six light bulbs are arranged in a circle. Three consecutive bulbs are chosen uniformly at random and turned on, while the other three remain off. The bulbs are then covered, so you cannot see which ones are on. The game consists of six rounds. In each round, you choose one unprocessed bulb and either inspect it or predict whether it is on or off. If you inspect a bulb, you immediately learn its state, but earn nothing for that bulb. If you make a prediction, the bulb remains covered and you do not find out whether your prediction was correct until the end of the game. You earn $1 for each correct prediction. Under the optimal strategy, what is your expected payoff? Post your answers in comments.