The Sleeping Beauty Problem: How To Really Solve It?

The Sleeping Beauty problem is a decision theory puzzle that deals with the logic and uncertainty of experience. Arnold Zuboff originally proposed it in the 1980s. Later on, people such as Robert Stalnaker and Adam Elga popularised the problem.

The problem features a simple thought experiment involving Sleeping Beauty (the fairy tale princess). Beauty agrees to undergo a special decision theory experiment under controlled conditions. You and I are the professionals conducting the experiment.

The experiment begins on Sunday when you put Beauty to sleep using a special sleeping pill. Immediately after she falls asleep, I toss a fair coin. If the flip results in ‘heads’, we would wake Beauty up on Monday morning and conclude the experiment.

If the flip results in ‘tails’, we would still wake her up on Monday morning. But immediately after she wakes up, you would once again give her the special sleeping pill. After she falls asleep again, we would wake her up on Tuesday morning and conclude the experiment.

In either case, the special sleeping pill wipes her memory out, and each time she wakes up, we brief her about the details of the experiment.

Given this setting, each time Sleeping Beauty wakes up, we ask her (after briefing her) what she thinks is the probability of the coin having landed ‘heads’. What do YOU think Beauty’s answer should be for it to be correct?

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The Intuitive Approach to the Sleeping Beauty Problem

The answer to this problem is straightforward! Regardless of how many times Beauty undergoes her sleep/wake cycles, it does not change the probability of a fair coin toss. It is ½ or 50%.

At least, that is what our intuitive understanding of probabilities leads us to think. So, what is the big deal with this puzzle then? Well, the challenge lies in the epistemic background for this problem.

If you are a fan of Bayesian probability, you would need to take the conditional probabilities of evidential experiences into account. In other words, Beauty has to consider the probability of the coin having landed ‘heads’ GIVEN other events at play.

This approach, of course, is not so intuitive. Let us see where it leads us.

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The Counter-intuitive Approach to the Sleeping Beauty Problem

Given the fact that Beauty loses her memory every time she wakes up, she would not know whether it is a Monday or a Tuesday. Therefore, the probability that the current day is a Monday or a Tuesday, given that the coin landed ‘tails’ would be the same:

Based on the Bayesian formula for conditional probability, we know the following:

When we equate these two expressions, we get the following result:

For ease of reference, let us label this result as equation A. Given this situation, if Beauty “assumes” that it is indeed a Monday, the probability of the coin landing ‘heads’ given it is a Monday is the same as that of the coin landing ‘tails’ given it is a Monday.

This is a perfectly valid assumption to make because Tuesday might or might not come to pass. But Beauty can be sure that Monday WILL come to pass. As before, we could use the Bayesian formula for conditional probability to arrive at equation B as follows:

When we compare equations A and B, we see that these three probabilities are equal:

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The Counter-intuitive Solution

We just figured out from equation C that three events from our experiment have an equal probability of occurring. If you consider the range of ALL possibilities in this experiment, these are the ONLY possibilities:

Since the sum of all probabilities (in our experiment) equals 1, the counter-intuitive solution is that each probability equals one-third (3x = 1):

Controversy Involving the Sleeping Beauty Problem

Until this day, the sleeping beauty problem is heavily disputed. One set of experts argues that the probability of ½ holds. These people argue that since Sleeping Beauty never learns any new information after her memory is wiped, the problem remains in the Frequentist domain.

Another set of experts argues that since she learns the nature of the experiment each time she wakes up, she learns new evidence which moves the problem to the Bayesian domain. I see myself quite strongly in this camp.

Although Beauty cannot know whether it is a Monday or a Tuesday, she can be sure that it is NOT a Sunday (from her knowledge of the experiment setup). 

This prior information has to be taken into account. And the Frequentist approach does not allow for that, whereas the Bayesian approach does!

Having said this, I acknowledge that the issue is by no means settled and finalised. The arguments can get deeply technical around this topic. What do you think is the correct answer/approach?

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References: Arnold Zuboff and Adam Elga.

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Further reading that might interest you:

• How To Really Understand The Philosophy Of Inferential Statistics?
• How Easy Is It Really To Predict The Future?
• Correlation (II): How To Spot The Surrogate Endpoint Problem?

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