You've been approached to assist in the furtherance of SCIENCE as a research subject at the Institute of Mad Science (local branch)!
The experiment you'll be involved with is called the Sleeping Beauty Study. The informational pamphlet you were given explains that, on Sunday, you'll be given a drug that will put you to sleep. During the study, the researchers will wake you up either once or twice, give you a brief quiz, and then put you back to sleep. The drug they use for knocking you out also erases your short-term memory, so you'll have no way of knowing if you're being woken up for the first time or the second time.
They'll decide how many times to wake you by flipping a coin. If it comes up heads, they'll only wake you on Monday. If it comes up tails, they'll wake you on both Monday and Tuesday. In either case, they'll you wake up on Wednesday (without quizzing you) to tell you the results of the experiment and send you home.
To motivate you to participate, you're being offered a cash prize of \$1000. If you answer every question correctly, you get the whole prize. If you answer every question incorrectly, you get nothing. If you get only some of them correct, you get half the prize, \$500.
The quiz will consist of one question: "Did the coin come up heads or tails?"
What strategy can you choose ahead of time that will maximize your expected winnings?