Sleeping Beauty Study at the Institute of Mad Science [closed]

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?

• @Deusovi I had thought that this one would be on-topic because A) rot13(lbh qba'g arrq gb fbyir gur haqreylvat (snzbhf) zngurzngvpf ceboyrz va beqre gb fbyir gur chmmyr), and B) rot13(ng yrnfg gb zr, gur erfhyg jnf harkcrpgrq be pbhagrevaghvgvir). The latter of those is even spelled out in the page you link to as one of the things that could make something on-topic. But if other people agree that it's off-topic, I can remove it. – Admiral Jota Aug 23 '19 at 18:47
• Both of those statements are true, but not particularly relevant - the problem is solved with straightforward and routine calculation, so it's off-topic according to our "textbook-style" close reason. – Deusovi Aug 23 '19 at 18:49
• I looked a bit deeper than the straightforward and routine calculation bit, and got surprised enough to vote for reopening. – Bass Aug 23 '19 at 23:42

The Sleeping Beauty problem itself is a famous problem in the philosophy of probability, and obviously we aren't going to resolve it here. Fortunately, the question here is more concrete, so let's just do it.

Another way of describing your payoffs in the two-awakening (tails) case is: you get \$500 for each correct answer. Your expected gain is the average of the expected gains in the (equiprobable) "heads" and "tails" cases. Because of the amnesia, your expected gain is the same on each day in the "tails" case, so your expected gain is twice that for a single day. Hence, in the "heads" case you get \$1000 if you say heads; in the "tails" case you get (in expectation) \$1000 if you say tails. The average of (\$1000 if you say heads, else 0) and (\$1000 if you say tails, else 0) is \$500 regardless of what you say.

Of course, you need not say the same thing every time.

You might, e.g., decide to pick randomly. But you will make your choice the same way every time, because of the amnesia (and because the researchers avoid letting you know when it is or which way the coin landed). So there will be some probability $$p$$ that you pick heads and it will be the same probability in all three situations, and then your final expectation will be $$p$$ times the expectation if you pick heads plus $$1-p$$ times the expectation if you pick tails -- and since those are the same, the final expectation doesn't depend on $$p$$.

Conclusion:

You can adopt whatever strategy you like, and your expected winnings will be exactly \\$500. (If you were somehow able to adopt a strategy that makes your choice correlate somehow with which way the coin toss went, then things would be very different. But the setup is designed to prevent this.)

• should this not be spoilered? – Ben Barden Aug 23 '19 at 17:06
• Maybe my explanation wasn't as clear as it could be, but I very much wasn't assuming that "pick heads" and "pick tails" are the only possible strategies. On reflection, my answer's pretty unclear on that point so I'll say some more. (An earlier version of my answer, which I didn't post, was more explicit but uglier. I'll see if I can manage explicit but not ugly.) – Gareth McCaughan Aug 23 '19 at 23:44
• I didn't spoiler my answer because the calculation's routine enough that there didn't seem much point. But since clearly at least one reader prefers it spoilered, spoilered it shall be :-). – Gareth McCaughan Aug 23 '19 at 23:51
• Anyway, @Bass, I've said a bit more about strategies other than "pick H" and "pick T". I think the answer is more or less back to its original level of ugliness, but never mind :-). – Gareth McCaughan Aug 23 '19 at 23:52