Sleeping Beauty Problem Divides Mathematicians; Shows Belief Can Lead To More Than 1 Rational Conclusion

The Sleeping Beauty problem that became popular in 2000 still keeps mathematicians awake. Apparently, they couldn't find a clear answer to the problem.

What Is the Sleeping Beauty Problem?

In mathematics, there are always straightforward answers, especially when the problems are simple. But there is still no unanimous agreement on the Sleeping Beauty problem.

Experts in philosophy and mathematics have divided into two groups and cited arguments for each group nonstop, frequently quite convincingly. There are more than 100 technical articles on this conundrum, and practically everyone who hears about the Sleeping Beauty thought experiment forms their own strong view.

Experts are baffled by the following issue - Sleeping Beauty consents to participate in a study. She receives a sleeping medication on Sunday and passes out. The experimenter throws a coin after that.

On Monday, if "heads" is called, Sleeping Beauty will be roused. Then they give another sleeping medication.

If "tails" appear, Sleeping Beauty is awakened on Monday, put back to sleep, then awakened once more on Tuesday. She then receives a second sleeping pill.

On Wednesday, they wake her again in both scenarios, and the experiment is over.

The crucial factor in this situation is that Sleeping Beauty has lost all memory of previous awakenings due to sleeping medication. She can't tell if it's Monday or Tuesday when she wakes up. Sleeping Beauty is not informed by the experimenters of the day's or the coin toss's outcome.

But each time she wakes up, they ask her the same question: What are the chances the coin will land on its head?

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Different Probabilities Explained

Theoretical physicist Manon Bischoff, an editor at the Scientific American partner newspaper Spektrum, had the first insight that Sleeping Beauty should predict 1/2. The likelihood of the coin falling on heads or tails is always 50%, independent of the outcome of the rest of the experiment.

When he became aware of the issue, American philosopher David Lewis had the same opinion. After all, before putting Sleeping Beauty to sleep, one could even toss a coin. She has no further information about the circumstance due to how the experiment was designed. Thus logically, she should answer that the chance is 1/2.

However, there are convincing justifications for a chance of 1/3. If you consider Sleeping Beauty's experience, one of three things could happen:

• She wakes up Monday, and heads was thrown.
• She wakes up Monday, and tails was thrown.
• She wakes up Tuesday, and tails was thrown.

What are the chances that each event will occur? This can be studied mathematically and empirically. Imagine flipping a coin 100 times and getting heads 48 times and tails 52 times. In other words, the Monday/heads scenario happens 48 times, while the Monday/tails and Tuesday/tails scenarios each happen 52 times.

Because Monday/tails always come after Tuesday/tails, the odds of all three events are equal. Hence the answer must be 1/3. This means that when Sleeping Beauty is woken and asked about the likelihood of the coin flip is head, she should respond with 1/3.

There are some intriguing uses for this conundrum. It can be used to think widely about probability and decision-making by mathematicians and philosophers.

This hypothetical example demonstrates how someone's beliefs-in this case, those of Sleeping Beauty-can result in more than one logical conclusion. It also emphasizes the distinction between the range of potential outcomes of an experiment (such as flipping heads or tails) and the range of possible participant experiences.

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