The Sleeping Beauty Puzzle Has Two Contradictory Correct Answers

January 19, 2016 | Elizabeth Knowles

Mixed media cutouts of sleeping beauty
Photo credit: Melanie Hughes/Flickr (CC BY 2.0)

The mathematical community is divided on this one.

The sleeping beauty problem is pretty simple. You don’t need much of a math background to solve it. However, there are two possible solutions you may come up with, and no matter how convinced you are that you are right, many other people will be convinced that you are wrong.

Here’s the puzzle: Sleeping beauty loves science so she agrees to participate in an experiment. The scientist tells her that he is going to put her to sleep on Sunday and then flip a coin. If it comes up heads, he will wake her up on Monday. If it comes up tails, he will wake her up on Monday, but then put her back to sleep, administer an amnesiac that will make her forget that she woke up, and wake her up again on Tuesday.

digram showing the two possible scenarios. A) Heads. She wakes up Monday. B) Tails. She wakes up Monday, forgets and wakes up Tuesday.

So, after listening to what the scientist says, sleeping beauty goes to sleep. Before she knows it, he is waking her up. He asks her a simple question: What is the probability that the coin came up heads?

There are two ways to think about this problem:

First, you could assume that the coin is fair, which means that there is a ½ probability that the coin came up heads. That’s just the way coin flipping works. She could have answered the question before he even put her to sleep. She knew that regardless of how the coin fell, he would wake her up so she has no new information and has no reason to change her answer.

The second way to think about the problem leads to the probability of heads coming up being ⅓. Since she doesn’t know what day of the week it is, there are three possible situations she could be in. It could be Monday and the coin came up heads, in which case she will get to stay awake. It could be Monday and the coin came up tails, in which case she will be heading back to sleep very soon. Or, it could be Tuesday and he could be waking her up for the second time after the coin came up tails. In these situations, the coin came up heads for 1 out of 3 of them.

In this second way of thinking about the problem, if the scientist repeated the experiment many, many times, and Sleeping Beauty guessed “heads” every single time, she would only be right one third of the time.

Both solutions have valid arguments, but are completely contradictory! Are you on team ½ or team ⅓? Let us know in the comments!

If you liked this brain teaser, try out our other logic puzzles!

Editor's note (January 19): The explanation of the second solution has been edited for clarity.

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