The Sleeping Beauty Problem: Revisited

In a Nutshell

Here’s a thought experiment with an interesting twist. I won’t give that away just yet.

It goes like this.

On Sunday you go to sleep. Someone in the same room now tosses a coin.

If it lands heads, you are woken the next day and interviewed. If it lands tails, you are woken the next day, asked a question, you go straight back to sleep and then the same happens on the Tuesday. Now, whenever you wake, you don’t know whether it is Monday or Tuesday and you don’t even know whether you have been woken before. It’s all a bit of a haze. In the original version of this problem, you are given a memory-erasing drug but let’s not over-complicate things.

You are Sleeping Beauty.

The question you are asked each time you wake is a simple one:

What probability should you now assign to the coin having landed heads?

You might think the answer is obvious. It was a fair coin when you went to sleep and nothing has happened to change that. Heads and tails each had a probability of 1/2. So it’s still a half — yes?

Halfers and thirders

There are two camps here — halfers and thirders.

The halfers say 1/2. Sleeping Beauty knew before going to sleep that she would certainly be woken at least once, so waking up has apparently given her no new information about the result of the toss. So it’s still a half.

The thirders see things differently.

Whenever Beauty wakes, she could be in one of three situations:

1. Heads and it’s Monday.
2. Tails and Monday.
3. Tails and Tuesday.

Only one of these involves heads. So, the probability of heads should be 1/3.

Both arguments seem reasonable but they can’t both be right.

Would you bet on it?

One way of thinking about the problem is to turn it into a bet. Suppose that every time Sleeping Beauty wakes she is offered odds of 2 to 1 that the coin landed heads. In other words, she stakes £10 and wins £20 if heads occurred. So, if the coin landed heads, she wakes once, places one £10 bet and wins a net £20. If it landed tails, she wakes twice and loses £10 on each occasion. Her total loss is £20.

Since heads and tails are equally likely when the coin was first tossed (we assume the coin isn’t rigged), over time the £20 gains and £20 losses balance one another.

So 2 to 1 looks like fair odds. And odds of 2 to 1 correspond to a probability of 1/3.

That seems a pretty good argument for the thirders.

What has Sleeping Beauty learned?

Before going to sleep on Sunday, Beauty knows that the coin is fair. The probability of heads is therefore 1/2. But she knows in advance that if the coin toss is heads, she will wake just on Monday. If tails, she will wake on Monday and Tuesday.

So the halfer asks: what new information has she actually received that should make her change her mind about the coin?

The thirder has an answer. Beauty may not have learned anything unexpected, but she now knows that she is experiencing one of the awakenings generated by the experiment. Tails generates twice as many such awakenings as heads. From the point of view of a randomly encountered awakening, tails should therefore be twice as likely.

And there, I think, lies the difficulty.

Are we asking about the probability that a fair coin lands heads? Or are we asking about the probability that she is waking up in a timeline generated by the coin landing heads? Remember that she is twice as likely to be waking up on a Tails day (Monday and Tuesday) than a Heads day (just Monday).

The first question seems to point to 1/2.

The second seems to point to 1/3.

Are these really two different questions, or are there two ways of reasoning about the same question? The jury is still very much out.

So what should Sleeping Beauty say when she opens her eyes?

I think I would say 1/3. But that’s because I’m a betting man, and I believe in reason.

Sleeping Beauty, on the other hand, is no fool and she knows that the coin was tossed just once, and it’s a fair coin. Who can blame her if she says 1/2, and sticks with it. And that’s the twist!