The Sleeping Beauty problem

2023-11-23

I recently came across the Sleeping Beauty problem, thanks to this excellent Veritasium video, and I thought I’d share some of my thoughts as someone completely unqualified to do so. The problem goes something like this:

Suppose that Sleeping Beauty is put to sleep on Sunday, and she is to be awakened some number of times the following week before returning to eternal slumber (ignoring any chance encounters with a prince that may ruin our experiment). Each time she wakes up, she will have no memory of previous awakenings, if any occurred. A fair coin is flipped to determine the number of times to wake her: if the coin lands on heads then she will be woken only once, on Monday, and a tails means she will be woken twice, on both Monday and Tuesday. When awoken, a researcher will ask her what the probability is that the coin came up heads. What should she say?

There are two answers that most people will be drawn to: either 1/2 or 1/3, with proponents of these answers being known as Halfers and Thirders respectively. The Halfers argue that a fair coin has a 50% chance of coming up heads, and so this should clearly apply to the answer given by Sleeping Beauty. Conversely, the argument from the Thirders is as follows: Sleeping Beauty will be woken up three times, twice when the coin came up tails and once when it came up heads, so the probability is one out of three.

Now, I didn’t study Philosophy, and while I do have a Maths degree it involved no Probability - I chose only pure Maths modules, which are widely agreed to be both “cooler” and vastly less practical when it comes to finding gainful employment. Ultimately, I think that this question comes down to semantic differences between the usage of probability between the Halfers and Thirders, so I’d argue that answering this question requires us to loosely define probability, and the answer will fall out naturally.

To me, the probability of an event can be thought of as the odds I’d want were I to bet on that event happening while expecting to be better off than had I not made the bet. For instance, if someone is about to roll a dice then I’d be happy to make a bet with them that the dice would come up a 6 provided that they would give me at least 6 times the amount I’d wagered if I won. Therefore, I’ll reframe the question in terms of a financial decision. I assert that the Sleeping Beauty problem is essentially equivalent to the following:

The researchers still flip the coin on Sunday, but they then leave poor Sleeping Beauty alone and instead spend their time approaching people randomly on the street, offering them a bet on whether the coin they flipped the previous Sunday came up heads or tails. If it came up heads, they approach just 1 person, and if it came up tails then they approach 1,000 people (updating the ratio from 1:2 to 1:1000 to help illustrate the point). We make all of the obvious assumptions to match this problem to the original one, i.e. that none of these people have any knowledge of who else has been approached and do not see any other approaches. However, the people are told about the experiment, so they do know that only 1 person is approached if the coin comes up heads and that 1000 are approached in the tails case.

Suppose you were among those approached. Which would you choose?

I would choose tails every time, and I think most rational people would too. Indeed, if this experiment was repeated 10 times and all participants followed this logic, we’d expect that 5000 of them would be better off and only 5 would leave with nothing, with the entire group becoming wealthier in total provided that they accepted odds of greater than 1:1000 (that is, odds ensuring that a wager of £10 would yield more than a penny in winnings if correct). Translated back to the original problem, Sleeping Beauty would be better off choosing tails and should be willing to bet on this fact with odds of at least 1:2. Given the chance, I’d take the money rather than arguing over semantics.