Hey folks,

here's an interesting puzzle that I've been thinking about lately. It's related to Bayesian probability theory and self-location. The interesting thing about it is that there seems to be no consensus on what the correct answer is, even though the problem is very simple to describe.

The wording from Wikipedia is as follows:

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin is then tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday and Tuesday. But when she is put to sleep again on Monday, she is given a dose of an amnesia-inducing drug that ensures she cannot remember her previous awakening. In this case, the experiment ends after she is interviewed on Tuesday.

Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"


So... what do you think?

Is there an answer?.. she can't possibly know, she has no frame of reference unless she's allowed access to a calender.

Seems like a pointessly contrived quandry to me.

Edited... It didn't need the quote.

Yes, the problem is contrived, but it reveals an interesting paradox. Basically, there are only two possible answers, it's either 1/2 or 1/3. People who have thought about this problem are somewhat evenly divided between the two.

It's not a pure logic puzzle either, it has some interesting philosophical implications regarding the anthropic principle and cosmology.

Well considering that she has no frame of reference then anything she says is a guess.

So one answer, right or wrong, is as probable and relevent as the other, to her.

So that being the case, there is no answer, for her, it's all the same.

An answer would only be relevent to an outside observer, and then they would know the answer anyway.

I think she's going to be under the impression the coin came up heads. She's always going to be awakened on Monday. However, there's a 50% chance that she'll also be awakened on Tuesday. Yet, should that occur, she won't have the ability to recall having been awakened once before. Therefore, she'll think it's Monday no matter what.

It reminds me of a puzzle with a hostage, a set of doors (one for escape and one for death), and two guards (one instructed to only lie, one instructed to only speak truth), or something like that.

Seems similar to Monty Hall. So far I have tried to convince myself of either side and find myself equally convinced... Guess this is why it could be considered a paradox.

I don't mean to be rude or dismissive, but as torment fan noted, there isn't an answer. Whatever she said, it would be a complete guess.

I also don't really see how it's a paradox. What on earth does this have to do with cosmology or philosophical implications?

You could simplify it by saying: If you give someone anesthetic and ask them to guess how long they were asleep for, will they guess the right answer..... I don't see how that's such a thought provoking question.

Superficially yes, but the Monty Hall problem has one unambiguously correct solution. It's a bit confusing initially, but once you understand it there's no question about it anymore.

The Sleeping Beauty problem, however, seems to have two different apparently correct but mutually exclusive solutions.

You said it... seems like a lot of brain bending over a nothing.

I'm starting to get it now... The Monty Hall Paradox... Take three doors, with a prize behind one. Chance of winning: 1/3. But, when the player picks a door, a losing door that went unchosen is opened. The player is then given the option to switch from the door they picked to the remaining door. Most players stick with what they chose, incorrectly thinking two doors left = 1/2 chance to win...

BUT: There are still three doors! The open door simply has 0/3 chance to win. The door the player chose has 1/3 chance, and the other has 2/3 chance to win. And therein lies the paradox.

Edit: To Monty Hall, at least. Can't wait for an explanation to this Sleeping Beauty paradox!

Of course, in real life with Murphy's Law fully in effect on the first take the coin lands on the edge. On the second take the Sleeping Beauty wakes up on Tuesday with erased memory not only of Monday's awakening, but of volunteering and instructions as well, calls the police and gets the researchers convicted for kidnapping.

I think there's some confusion about the meaning of the word "credence." Basically, it means the following: From Beauty's point of view, which probability should she assign to the hypothesis that the coin landed heads? What it doesn't mean is: What side did the coin land on? That, of course, she cannot answer.

Basically, you can argue in two different ways:

1. "1/2 position". The coin is tossed before Beauty is put to sleep. She doesn't know which side the coin landed on, so the can only assign a probability of 1/2 to heads. Right after she is woken up, she has gained no relevant additional information, so her credence of the probability of heads should still be 1/2.

2. "1/3 position." You can argue for this position based on long-run statistics. Imagine you repeat the experiment 1,000 times. Then the coin is expected to land heads 500 times and tails 500 times. Beauty would be awoken 500 times on Monday after the coin landed heads, 500 times on Monday after the coin landed tails, and 500 times on Tuesday after the coin landed tails. So she would be awoken 1,500 times, but only in 500 cases would the coin have landed heads. That is, in 1/3 of her awakenings the coin had landed heads, so her credence for heads should be 1/3. (You can also argue for this position in a more analytic way that doesn't involve repetitions.)

Regarding the philosophical implications, I'd like to get into those a little later since they require more explanation.

Ahh, ok. I stand by the 1/2 position. More awakenings from a tails landing doesn't change the toss-up probability. The coin will always have two sides.

While my research of this issue is limited, I'd love to share with you the marvelous Placek's Paradox (also known as Kraków-Warsaw Paradox) which happens to undermine the theory.

Individually it may seem that you have a 1/2 chance of it being heads, but in the long run it is always going to be twice as likely to be tails. This is in some ways similar to Monty Hall - with the MH problem, you are statistically more likely to win if you swap after the first box is open, but only in long run stats. Individually, your first guess is as good as your second, as you may have picked correctly.

So, in both cases, individually it may as well be a guess (1/2 in SB, 1/3 in MH), but statistically speaking there is a "correct answer".

In the case of Sleeping Beauty, she is specifically asked for a statistical probability of the coin having landed heads, therefore the answer is always 1/3.