The Sleeping Beauty Problem

Scott Hardie | February 22, 2023
Kelly Lee and Matthew Preston were both halfers from the start and never wavered.

Me, I was a thirder at first and quite confident about it, thanks to probability puzzles like The Monty Hall Paradox that demonstrate some biases in our understanding of probability and how something with a truly random outcome can subtly become less than random. But gradually, I came around to being a halfer after watching the video again and discussing it with Kelly and Matthew.

It seems to me that the conundrum lies in how strictly you interpret the question's wording:

She'll be asked one question: "What do you believe is the probability that the coin came up heads?" How should she answer?

Asked this way, the answer should always be one half, because that's always the probability of heads in any of the possible outcomes:
• If the coin toss came up heads and today is Monday, the probability that the coin toss came up heads was one half, so that's what Sleeping Beauty should answer.
• If the coin toss came up tails and today is Monday, the probability that the coin toss came up heads was one half, so that's what Sleeping Beauty should answer.
• If the coin toss came up tails and today is Tuesday, the probability that the coin toss came up heads was one half, so that's what Sleeping Beauty should answer.

The crux of the thirder argument is that when Sleeping Beauty wakes up...

...she learns that she's gone from existing in a reality where there are two possible states (the coin came up either heads or tails) to existing in a reality where there are three possible states (Monday heads, Monday tails, or Tuesday tails). And therefore, she should assign equal probability to each of these three outcomes.

But that's only prudent if the three possible outcomes are all equally likely, and they are not! Sleeping Beauty is not being asked, "if all possible ways that you could be asked this question were equally likely, what is the probability of the circumstances that led to this question?" One third would be the correct answer to that question. As one of the video's commenters succinctly put it, the odds of winning the lottery are not 50-50 just because there are two possible outcomes.

There may be something that the Veritasium video didn't explain about mathematical probability, some reason beyond my layperson's understanding why "therefore" in the above quote makes logical sense. But in lieu of that, treating all three scenarios as equally likely not only seems wrong, it also leads to all kinds of absurd conclusions in the rest of the video, such as the pen-and-paper experiment. Of course that chart is going to wind up with a roughly equal number of tally marks in each of the three columns, because the two tails columns will both get a mark if tails is flipped! A tally of all possible scenarios is not relevant information; instead it's better to write down what Sleeping Beauty's answer should be in each possible scenario, like I did above in three bullet points. Especially dumb is the "million black marbles and one white marble" thought experiment, which again answers a different hypothetical question than the one asked; a more relevant wording would be, "given that a coin toss resulted in either one white marble for heads or a million black marbles for tails, and you are about to draw one marble from the bag, what is the probability that you pull out a white marble," which would be one half.

I like Veritasium because the channel is smart and interesting, but it's way off base this time. Do you agree or disagree?