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The Sleeping Beauty Problem

March 4, 2017

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening.

A fair coin will be tossed on Sunday evening after she is put to sleep, to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed, she is asked, “What is your belief now, as a percentage, in the proposition that the coin landed heads?”

What should Beauty’s answer be?

To one way of thinking about this, the answer is clear. The coin was tossed once prior to her awakening, however many times she is woken, whether once (if it landed heads) or twice (if it landed tails).

Since the fair coin was tossed just once, and no further information is obtained by Beauty at the time she is awoken and interviewed, the answer she should give should be 50 per cent, i.e. a 1 in 2 chance that the fair coin landed heads.

To another way of thinking about it, she is interviewed just once if it landed heads (on the Monday) but she is interviewed twice if it landed tails (on Monday and Tuesday). She does not know which day it is when she is woken and interviewed but from her point of view there are three possibilities. These are:

  1. It landed heads and it is Monday.
  2. It landed tails and it is Monday.
  3. It landed tails and it is Tuesday.

So there are three possibilities, of equal likelihood, and two of these involve the coin landing tails and just one for the coin landing heads. So the answer she should give should be 33.3 per cent, i.e. a 1 in 3 chance that the fair coin landed heads.

So which answer is correct? The world of probability is by and large divided into those who are adamant that she should go with ½ (the so-called ‘halfers’) and those who are equally adamant that she should go with 1/3 (the so-called ‘thirders’). Are they both right, are they both wrong, or somewhere in between?

A way that I usually advocate to resolve seemingly intractable probability paradoxes is to ask at what odds Beauty should be willing to place a bet.

So, if in this experiment Beauty is offered odds of 1.5 to 1 that the coin landed heads, should she take those odds? If the correct answer is a half, those odds are attractive as the correct odds should be 1 to 1 (evens). If the correct answer is a third, those odds are unattractive as the correct odds should be 2 to 1.

So what should Beauty do if offered odds of 1.5 to 1? Bet or decline the bet?

The simplest way to resolve this is to ask what would happen if she accepted the odds of 1.5 to 1 and placed a bet of £10 each time. When the coin came up heads, she would be awoken just once, placed the £10 bet and won £15. However, when the coin landed tails she would be awoken twice and placed two bets of £10, i.e. a total of £20 and lost both bets.

So her net outcome of this betting strategy would be a loss of £5.

This suggests that a half is the wrong answer as to the probability that the coin landed heads. At odds of 2 to 1, on the other hand, she would place £10 on the one occasion she would be awoken, i.e. Monday, and would win £20. However, when the coin came up tails, she would lose £10 on the Monday and £10 on the Tuesday, i.e. £20. Her expected outcome would in this case be to break even. This suggests that odds of 2 to 1 are the correct odds, which is consistent with a probability of 1/3. Some ‘Halfers’ argue that Beauty should be assigned a chip of half the value if the coin lands Tails than if it lands Heads, although she will be unaware of the value of the chip when she stakes it. In this case, she would indeed break even by betting at even money odds, but there seems no reasonable case to be made for applying this arbitrary fix to the experiment.

Applying the ‘betting test’ to this problem, therefore, suggests that Beauty’s answer when she is woken up should that there is a 1 in 3 chance that the coin landed heads when tossed after she was put to sleep on the Sunday.

But how can this be right, when the fair coin was tossed just once, and we know that the chance of a fair coin landing heads is ½? If this is the ‘prior probability’ Beauty should assign to the coin landing heads, and she is given no further information about what happened to the coin when she is woken and questioned, on what grounds should the probability she assigns change? The only information she acquires is that she has been woken and questioned, but she knew that would happen in advance, so this is not new information. Given she assigns a prior probability of ½ to the coin coming up heads, and she acquires no new information, it is perhaps difficult to see on what grounds she should change her opinion. The posterior probability she assigns (after she acquires all new information) should be identical to the prior probability, because she has acquired no new information after being put to sleep to change anything.

This is the kernel of the conundrum, and it is why there is a long-standing and ongoing debate between fervent so-called ‘Halfers’ and ‘Thirders.’

So the question is whether there is a correct answer, and that one school of thought is simply wrong, or whether there is no correct answer and both schools of thought are wrong or only right under one interpretation of the question.

It seems to me that there is, in fact, a straightforward answer, which resolves the problem. To see this, we need to identify the actual ‘prior probability’ that the coin tossed after Beauty goes to sleep is Heads.

This depends on the question we are seeking to answer, and what information is available to Beauty before she goes to sleep.

If she is simply told that a coin will be tossed after she goes to sleep, and nothing else, then her correct estimate that the fair coin will land on heads is ½. This is the answer to a simple question of how likely a fair coin is to land Heads with no conditions, i.e. the unconditional probability that the coin will land Heads is 1/2.

If she is given the additional information, however, that she will be woken just once if the coin lands Heads but twice if it lands Tails (albeit she will remember just one of the awakenings), then we are posing a very different question.

The new question she is being asked to answer is to estimate the probability that whenever she awakens, that her awakening resulted from the coin toss landing Heads. Since she has just one awakening when the coin lands Heads, but two awakenings when it lands Tails, the probability that any particular awakening occurred from a Heads flip is 1/3, i.e. the conditional probability that the coin landed Heads given any particular awakening is 1/3.

By extension, if she is told she will be woken 1,000 times if the coin lands Tails but only once if the coin lands Heads, then her correct estimate of the probability that any particular awakening resulted from the coin landing Heads is 1/1001.

So the ‘prior probability’ Beauty should assign to the chance of a coin landing Heads after any particular awakening is actually 1/3 within the terms of the experiment, even before she goes to sleep. It is true that she has access to no new information whenever she awakens, but that simply means that her ‘prior probability’ of being awakened by a Heads flip remains at 1/3 after she is woken. This is totally consistent with Bayesian reasoning which states the prior probability of an event will not change unless there is new information.

Given, therefore, that she assigns a prior probability of 1/3 to any particular awakening arising from a Heads flip, this should be the answer she gives whenever she awakens, and also before she goes to sleep.

So the paradox resolves to the question Beauty is being asked to answer. What is the probability that a fair coin will land Heads? Answer = ½. What is the probability that whenever she is woken this awakening has resulted from a Heads flip? Answer = 1/3. She is consistent in these answers both before she goes to sleep and whenever she wakes. In other words, because Beauty knows that she will correctly answer 1/3 whenever she is woken, given the rules of the experiment, of which she is aware, she will answer 1/3 before she goes to sleep.

The resolution of the Sleeping Beauty Problem has implications for the so-called ‘anthropic principle’ more generally.

The ‘anthropic principle’ is the consideration that theories of the universe are constrained by the necessity to allow human existence, because our existence as conscious observers of the universe is a given. So any theory or model of the universe must have our existence as at least one possibility.

The simplest state of affairs would be a situation in which nothing had ever existed. This would also be the least arbitrary, and certainly the easiest to understand. Indeed, if nothing had ever existed, there would have been nothing to be explained. Most critically, it would solve the mystery of how things could exist without their existence having some cause. In particular, while it is not possible to propose a causal explanation of why the whole Universe exists, if nothing had ever existed, that state of affairs would not have needed to be caused. This is not helpful to us, though, as we know that in fact this Universe does exist.

In fact, we are faced with the fact that the positive and negative contributions to the cosmological constant cancel to 120 digit accuracy, yet fail to cancel beginning at the 121st digit. In fact, the cosmological constant must be zero to within one part in roughly 10120 (and yet be nonzero), or else the universe either would have dispersed too fast for stars and galaxies to have formed, or else would have collapsed upon itself long ago. How likely is this by chance? Essentially, it is the equivalent of tossing a coin and needing to get heads 400 times in a row and achieving it. Now, that’s just one constant that needs to be just right for galaxies and stars and planets and life to exist. There are quite a few, independent of this, which have to be equally just right, but this I think sets the stage. This is sometimes called the fine-tuning argument.

The parallel with the Sleeping Beauty Problem is that Beauty knows she has been awakened and so any explanation of this must have that awakening as at least one possibility, just as any theory of the Universe must have our conscious state as one possibility.

In terms of modelling the Universe, we might pose two possible theories. In one, all the physical constants we observe today are explained. They were designed that way or they have to be that way for some unknown reason. The second theory is that there could have been countless trillions of different ways that the physical constants could have arranged themselves, and only one of these is consistent with the Universe (and us) existing.

For simplicity of exposition, let us assume that the two theories are otherwise equal in terms of empirical evidence, scientific rigour, and so on, but the general point stands whatever.

In other words, from the perspective of an observer outside the Universe, these theories would be equally likely. Heads or Tails. ½.

But we as conscious observers of our existence are like Sleeping Beauty when she wakes. From our perspective, there is only one chance in countless trillions that we would be asking the question if the second theory is correct, which means from our ‘anthropic’ perspective the chance that the first theory is correct (the constants were designed that way or have to be that way) is trillions of times more plausible.

This has, of course, very important scientific, philosophical and theological implications, which demonstrates the power and importance of the Sleeping Beauty Problem as more than just a simple mind-bender.

Let us within this context now tackle the criticism of those who reject the larger importance of this vanishingly small possibility of the physical constants being randomly trillions to one in our favour on the grounds that if it wasn’t so, we would not have been around to even ask the question. This take on the ‘anthropic principle’ sounds a clever point but in fact it is not. For example it would be absolutely bewildering how I could have survived a fall out of an aeroplane from 39,000 feet onto tarmac without a parachute, but it would still be a question very much in need of an answer. To say that I couldn’t have posed the question if I hadn’t survived the fall is no answer at all.

Others propose the argument that since there must be some initial conditions, these conditions which gave rise to the Universe and life within it possible were just as likely to prevail as any others, so there is no puzzle to be explained.

But this is like saying that there are two people, Jack and Jill, who are arguing over whether Jill can control whether a fair coin lands heads or tails. Jack challenges Jill to toss the coin 400 times. He says he will be convinced of Jill’s amazing skill if she can toss heads followed by tails 200 times in a row, and she proceeds to do so. Jack could now argue that a head was equally likely as a tail on every single toss of the coin, so this sequence of heads and tails was, in retrospect, just as likely as any other outcome. But clearly that would be a very poor explanation of the pattern that just occurred. That particular pattern was clearly not produced by coincidence. Yet it’s the same argument as saying that it is just as likely that the initial conditions were just right to produce the Universe and life to exist as that any of the other pattern of billions of initial conditions that would not have done so. There may be a reason for the pattern that was produced, but it needs a more profound explanation than proposing that it was just coincidence.

A second example. There is one lottery draw, devised by an alien civilisation. The lottery balls, numbered from 1 to 49, are to be drawn, and the only way that we will escape destruction, we are told, is if the first 49 balls out of the drum emerge as 1 to 49 in sequence. The numbers duly come out in that exact sequence. Now that outcome is no less likely than any other particular sequence, so if it came out that way a sceptic could claim that we were just lucky. That would clearly be nonsensical. A much more reasonable and sensible conclusion, of course, is that the aliens had rigged the draw to allow us to survive, or else that the draw had to be that way because no other possible sequence of balls could physically emerge.

So the answer to the Sleeping Beauty Problem is 1/3 that she is in the Heads world if she is awakened once when the coin lands Heads and twice when it lands Tails. If awakened a million times in the Tails world but just once in the Heads world, the chance she awakes to a Heads world is 1 in a million and 1. The bigger question for humanity is what world we exist in, Heads (we have to exist) or Tails (there is effectively no chance that we exist). I call that the Possibility Problem, and it is a problem which would seem to have a probabilistic solution.



Using Bayes’ Theorem:

P (Heads I Wake up) = P (Wake up I Heads) . P (Heads) / P (Wake up)

If you adopt the Self-Sampling Assumption (SSA), you sample a person from within that world at random.

So, P (Heads I Wake up) = 1 . 1/2 /  1 = 1/2

If you adopt the Self-Indication Assumption, you take into account that you are more likely to exist in a world with more beings (or opportunities to experience) than in one with less. In this case, there are twice as many opportunities to experience waking up if the coin lands Tails than if it landed Heads.

So, P (Heads I Wake up) = 1 . 1/3 / 1 = 1/3

Further Reading and Links

Sections of this blog relating to the ‘anthropic principle’ and ‘fine-tuning’ have appeared in my related blog, ‘Why is there Something Rather than Nothing?’ Link at:

Bayes’ Theorem: The Most Powerful Equation in the World. Related blog.

Wikipedia entry on the Sleeping Beauty Problem

The Sleeping Beauty Problem. By Julia Galef. YouTube.

Philosophy- Epistemology. The Sleeping Beauty Problem. By Michael Campbell. YouTube.

Probably Overthinking It: A Blog by Allen Downey

Wikipedia entry on the Anthropic Principle

Wikipedia entry on Fine-Tuned Universe

Blog entry on The Vaughan Williams ‘Possibility Theorem’ and related applications.

Derek Parfit, ‘Why anything? Why this? Part 1. London Review of Books, 20, 2, 22 January 1998, pp. 24-27.

Derek Parfit, ‘Why anything? Why this? Part 2. London Review of Books, 20, 3, 5 February 1998, pp. 22-25.

John Horgan, ‘Science will never explain why there’s something rather than nothing’, Scientific American, April 23, 2012.

David Bailey, What is the cosmological constant paradox, and what is its significance? 1 January 2017.

David Albert, ‘On the Origin of Everything’, Sunday Book Review, The New York Times, March 23, 2012.

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