I recently learned of the Sleeping Beauty Paradox, I found it interesting so I thought I’d share.

Suppose you are part of an experiment: you will be made unconscious, and the experimenter will flip a coin. If it comes up heads, you will be revived at time t_1 and interviewed. If it comes up tails, you will be revived at time t_1, interviewed, then made unconscious again with your memory of being revived erased, and then revived again at t_2 and interviewed again.

Assume that there are no indicia of the passage of time, you only know the experimental setup and that you have just been revived and questioned; that you know the memory erasure to be completely effective and have no side affects; that you know the coin to be fair, etc.

You are revived by the experimenter, who asks you one question: “**What is your credence that I have flipped heads?**”

What is your answer?

The rest is hidden so you can consider before proceeding.

## Discussion

The Wikipedia article gives three possible answers:

**1/2** – If it’s a fair coin, there’s a 50-50 chance of flipping heads.

**1/3** – there are three possible revivals, and in only one of them would the experimenter have flipped heads.

**Mu** – the question is ambiguous. Wikipedia offers the following similar problem as a disambiguation:

Imagine tossing a coin, if the coin comes up heads, a green ball is placed into a box; if, instead, the coin comes up tails, two red balls are placed into a box. We repeat this procedure a large number of times until the box is full of balls of both colours. A single ball is then drawn from the box. In this setting, the question from the original problem resolves to one of two different questions: “what is the probability that a green ball was

placedin the box” and “what is the probability a green ball wasdrawnfrom the box”. These questions ask for the probability of two different events, and thus can have different answers, even though both events are causally dependent on the coin landing heads. (This fact is even more obvious when one considers the complementary questions: “what is the probability that two red balls were placed in the box” and “what is the probability that a red ball was drawn from the box”.)

I lean towards 1/2: There’s a 50% chance that it’s t_1 and the coin landed on heads; a 25% chance that it’s t_1 and it landed on tails, and a 25% chance that it’s t_2 and it landed on tails. I don’t see that being revived gives you additional information sufficient to change your credence that the coin was flipped to heads, your observations are completely consistent with either outcome.

But smarter people than me are convinced otherwise, and the paradox is part of the basis for some versions of the Simulation Hypothesis (because if you take the 1/3 position here, you should similarly think it’s more likely that you’re in one of millions of simulated worlds rather than the one real world).