In yesterday’s post, I described the so-called “Sleeping Beauty Problem”. Mathematicians and philosophers have been debating this problem for the past 15 years (most recently in response to an article at the Quanta Magazine site). What should Sleeping Beauty say when she wakes up on Monday or Tuesday and is asked the following question about a coin flip that happened on Sunday: “What are the odds that the coin we flipped on Sunday came up heads?”.
To review: Sleeping Beauty doesn’t know what day it is when she wakes up or whether she’s already woken up during the experiment. But she is told that if the coin flip on Sunday came up heads, she’s only being awakened on Monday. If, however, the coin came up tails, she’s being awakened on both Monday and Tuesday. “Halfers” think the odds that the coin came up heads is 1/2. “Thirders” think the answer is 1/3.
Here’s the answer Pradeep Mutalik, the author of the Quanta article, gave in response to a whole lot of comments:
This is the crux of the idea: that there are two propositions in the Sleeping Beauty problem:
1) the probability of heads when the fair coin is tossed, which is obviously 1/2. This is the proposition modeled by halfers.
2) the probability of heads encountered later, which thanks to the experimental conditions that are asymmetric between heads and tails, is 1/3. This is the proposition modeled by thirders.
… the same problem statement can mean tossed if understood one way and encountered if interpreted another way.
Once you accept that there are two valid interpretations, the solution of the Sleeping Beauty problem can be expressed in one sentence:
A fair coin is tossed: The probability of it landing heads when tossed is ½; the probability of Sleeping Beauty encountering heads later, thanks to the asymmetric experimental conditions, is 1/3.
And that, in a nutshell, is all there is to it.
Mr. Mutalik also added this note in response to my own comment on his article, in which I came out in support of the 1/3 answer:
As in my reply to Rich, your answer is correct for the heads “encountered” probability. The heads “tossed” probability is, and remains ½ until SB learns the actual result of the coin toss.
The point of all this is to show that there is no incompatibility in the thirder and halfer arguments. The same person can simultaneously be both a halfer and a thirder.
As a new supporter of the thirder position, I’m not convinced by Mr. Mutalik’s response. As I wrote in the previous post, it’s clear that the odds of getting heads or tails when a coin is flipped are always 1/2 heads and 1/2 tails. That’s how coin flips work (putting aside the extremely rare occasions when they land on their edge and don’t fall either way).
But the question presented to Sleeping Beauty is: “What are the odds that the coin flip came up heads?” Since she knows how the experiment is being conducted when she’s asked this question, shouldn’t she conclude that the odds are 1/3, not 1/2? After all, she knows that a coin that came up tails means this could be either Monday or Tuesday. If it came up heads, it can only be Monday. So the odds that the coin came up tails is twice as likely as the odds that it came up heads (i.e., 2/3 vs. 1/3).
It seems to me that asking Sleeping Beauty to ignore the information she’s been given about the experiment is asking her to give an answer she knows is incorrect. What a question means depends on its context. If the context in this case were simply: “Please explain how coin flips work. In particular, what are the odds that a fair, properly-executed coin flip will come up heads instead of tails, such as the coin flip we executed on Sunday?” In that context, the correct answer is obviously 1/2. Everybody should be a halfer when asked how coin flips work.
But that isn’t the context in which Sleeping Beauty is being asked to give an answer. She’s not being asked to explain how coin flips work in general. She understands that she’s involved in a bizarre, possibly unethical (the memory-destroying drug!) experiment and that the question she’s being asked is part of that experiment. She’s being asked what the odds are given her current situation. That’s why I think it’s more reasonable for Sleeping Beauty to be a thirder than a halfer. She should answer “1/3”. It’s twice as likely that the coin came up tails!
Whether she should be allowed to go back to sleep after giving the correct answer, so that she can later be awakened by a handsome young prince, or whether she should receive appropriate medical care and reacquire a normal sleeping pattern, is an ethical question, not a logical one, and beyond the scope of this post.