When Should We Rethink Probability? Exploring the Sleeping Beauty Problem
A Version of this article is published in my book, ‘TWISTED LOGIC: PUZZLES, PARADOXES, AND BIG QUESTIONS’. CRC Press/Chapman & Hall, 2024. https://www.amazon.co.uk/Twisted-Logic-Puzzles-Paradoxes-Questions/dp/1032513349
A Thought Experiment
The Sleeping Beauty Problem is a thought experiment that challenges our understanding of probability. It involves Sleeping Beauty, a coin toss, and a scenario where her memory is erased, leading to a debate between two main schools of thought: ‘halfers’ and ‘thirders’.
The Sleeping Beauty Experiment
The experiment plays out as follows: Sleeping Beauty participates in an experiment, starting on a Sunday. The course of the experiment depends entirely on the outcome of a fair coin toss. If it lands heads, Beauty will be woken and interviewed only on Monday. If it lands tails, she will be awakened and interviewed on both Monday and Tuesday. On each occasion, she is asked what chance she assigns to the coin having landed heads. After she answers, she is put back into a sleep with a drug that erases her memory of that awakening. The experiment in any case finishes on Wednesday, with Sleeping Beauty waking up without an interview.
In other words, Sleeping Beauty participates in a coin toss experiment. If the coin lands heads, she is woken and interviewed only on Monday. If tails, she is woken on both Monday and Tuesday, with each awakening followed by memory erasure. She is asked each time about the likelihood of the coin landing heads.
Probability Paradox: Halfers vs. Thirders
When presented with this experiment, two primary interpretations of how Sleeping Beauty should calculate the probability emerge. Halfers propose that since the coin is only tossed once and no new information is collected by Beauty, she should assert a 1 in 2 chance that the coin landed heads. On the contrary, Thirders argue that from Sleeping Beauty’s standpoint, there are three equally likely scenarios, two of which involve the coin landing tails and one after a heads. Specifically, these are:
- It landed heads, and it is Monday.
- It landed tails, and it is Monday.
- It landed tails, and it is Tuesday.
Therefore, Thirders suggest that whenever she wakes up, she should assign a 1 in 3 chance to the coin having landed heads.
The Betting Frame: Determining Fair Odds in the Sleeping Beauty Problem
One potential strategy for deciphering this complex issue is by considering it in terms of fair betting odds. For instance, if Sleeping Beauty were offered odds of 2 to 1 (£1 to win a net £2) that the coin landed heads, should she take the bet?
The best way to look at this is to think about what would happen if she accepted the 2 to 1 odds each time she woke up. If the coin toss results in heads, she’d be woken up once, bet £10, and profit £20. But if the coin lands on tails, she’d be woken up twice, place two £10 bets (a total of £20) and lose both times.
Her ‘average’ result with this betting strategy would be to break even. This implies that 2 to 1 represent the correct odds. These odds (£1 win to win a net £2) are consistent with a probability of 1/3. So, using this betting test, when Beauty wakes up, she should think there’s a 1 in 3 chance that the coin landed on heads. This supports the ‘Thirder’ case.
Shifting Probabilities: From Unconditional to Conditional
A critical step in unravelling this puzzle involves an examination of the ‘prior probability’. This is the probability assigned before the collection of any new information. If asked to estimate the likelihood of a fair coin landing heads without any additional conditions, Beauty should answer 1/2. However, with added information, the question can be reformulated into estimating the probability of her waking as a result of the coin landing heads. Here, thirders would argue for a 1/3 probability. So, what information does Beauty actually have when she goes to sleep that Sunday, and how does that affect the prior probability that she should assign to the coin landing heads? Bear in mind, though, that the coin is only tossed once, and it is a fair coin.
Conclusion: How the Sleeping Beauty Problem Combines Chance and Deep Thought
The Sleeping Beauty Problem is more than a statistical puzzle; it’s a probe into the nature of information and observation. It shows that our understanding of probability can significantly shift based on the framing of the question and the information available to us. Indeed, it shakes up how we think and makes us wonder about what ‘information’ really is. This serves as a powerful reminder that the real world, like the Sleeping Beauty Problem, doesn’t always have easy or clear-cut answers. The more we dig into this mind-bending problem, the more we learn from it.
Professor Leighton Vaughan Williams 
In Zuboff’s original version of the problem, SB is woken on each of the next trillion days, or on one randomly selected day among the next trillion, based on the flip of a fair coin. He does not mention which coin result means one, or one trillion. And the question is about whether she will be wakened once, or a trillion times.
Elga’s proposed problem simplified that to one, or two, wakings. Not days. It did not specify how the “one” was to scheduled. It did specify that Heads corresponds to one day, and so the question about Heads was equivalent to Zuboff’s.
To implement his solution, Elga added the Always-Monday, Tuesday-Tails schedule. But barring a valid objection, I will continue as if none of these differences affect the solution methodology. Because another variation, that similarly makes no difference to methodology, allows a trivial solution and points to the error made by one of the camps.
There are two possible coin faces (H and T), and two possible days (Mon and Tue) where a volunteer could be left asleep. With four possible combinations, we can implement the experiment with four volunteers, assign a different combination to each, and use each combination this way. Assignments can be secret, or we can tell each her combination with the provision that she can’t share this information.
On each day of the experiment, the one who was assigned the actual coin result, and the current day, is left asleep. The other three are brought together to discuss the probabilities that should be assigned to each, for the proposition that this is her only waking (Zuboff’s question), or that the coin landed on her assigned face (Elga’s question).
Regardless, they all know that the proposition is true for exactly one of the three awake volunteers. With no information to distinguish which it is, the probability must be 1/3 for each.
Here’s the error in the halfer argument – and most discussions, regardless of the answer that is favored. They assume, in Elga’s solution, that Tuesday does not happen if the coin lands on Heads. That it is not a part of any valid sample space. This is probably Elga’s fault, since his solution methodology was to carefully address different conditional probability problems where it was not a part of the condition. But my four-volunteer version forces it to be recognized as a valid outcome in the prior that is not included in the observation, since it is represented by an actual person who is missing.
In the more common version, once H+Tue is recognized as a valid outcome, the answer 1/3 is easier to see.