Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

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?”

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.

This, at least, is one seemingly reasonable way of looking at, and providing a solution to, the classic Sleeping Beauty problem.

Exercise

You volunteer to undergo the following experiment and are told all of the following details: On Sunday you will be put to sleep. Once or twice during the experiment, you will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes you forget that awakening.

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

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