Via Robert Wiblin here is a fun probability puzzle:
The Sleeping Beauty problem: Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking.
The puzzle: when you are awakened, what probability do you assign to the coin coming up heads? Robert discusses two possible answers:
First answer: 1/2, of course! Initially you were certain that the coin was fair, and so initially your credence in the coin’s landing Heads was 1/2. Upon being awakened, you receive no new information (you knew all along that you would be awakened). So your credence in the coin’s landing Heads ought to remain 1/2.
Second answer: 1/3, of course! Imagine the experiment repeated many times. Then in the long run, about 1/3 of the wakings would be Heads-wakings — wakings that happen on trials in which the coin lands Heads. So on any particular waking, you should have credence 1/3 that that waking is a Heads-waking, and hence have credence 1/3 in the coin’s landing Heads on that trial. This consideration remains in force in the present circumstance, in which the experiment is performed just once.
Let’s approach the problem from the decision-theoretic point of view: the probability is revealed by your willingness to bet. (Indeed, when talking about subjective probability as we are here, this is pretty much the only way to define it.) So let me describe the problem in slightly more detail. The researchers, upon waking you up give you the following speech.
The moment you fell asleep I tossed a fair coin to determine how many times I would wake you up. If it came up heads I would wake you up once and if it came up tails I would wake you up twice. In either case, every time I wake you up I will tell you exactly what I am telling you right now, including offering you the bet which I will describe next. Finally, I have given you a special sleeping potion that will erase your memory of this and any previous time I have awakened you. Here is the bet: I am offering even odds on the coin that I tossed. The stakes are $1 and you can take either side of the bet. Which would you like? Your choice as well as the outcome of the coin are being recorded by a trustworthy third party so you can trust that the bet will be faithfully executed.
Which bet do you prefer? In other words, conditional on having been awakened, which is more likely, heads or tails? You might want to think about this for a bit first, so I will put the rest below the fold.
There is a logical trap to watch out for.
- Since I will have no memory of any previous awakening, if I am awakened twice I will make the same choice both times (unless I am indifferent)
- So if I bet tails, and the coin has actually come up tails, I will be awakened twice, both time betting tails and winning. I would make $2. If it comes up heads I will be awakened only once and lose. I would lose $1.
- But if I bet heads I would win $1 if the coin comes up heads and lose $2 if tails.
If you accept 1-3, and you trust that the researchers tossed a fair coin you strictly prefer to bet on tails. In fact, you would only bet on heads if you were given 3 to 1 odds.
But you should not accept 2 or 3. On the other hand, 1 is absolutely true. I will make the same choice twice. But 2 and 3 make the fallacious jump to the conclusion that I must make the same choice twice.
Instead, when I am awakened, I am free to make either bet. And whatever choice I make now cannot have a causal effect on my choice in a (possible) second awakening. So I should think like this.
- In that other awakening I will make some bet, and my choice now has no effect on it.
- I should therefore measure the incremental gain or loss from my bet at this moment.
And we can solve the problem this way because it turns out that the incremental gain does not depend on how I will bet in the other awakening.
- Suppose that I would bet on tails in a second awakening. If I currently bet tails, then if the coin came up tails I will win twice, if heads I will lose once. Since the coin is fair, my expected payoff is 50 cents (.5 probability of winning twice, .5 probability of losing once.) On the other hand, if I currently bet heads, then if the coin came up tails I will win once, lose once; if heads I will win once. The expected payoff is again 50 cents. I am indifferent.
- Suppose that I would bet on heads in a second awakening. If I currently bet tails, then if the coin came up tails I will win once, lose once; if heads I will lose once. If I currently bet heads, then if the coin came up tails i will lose twice; if heads I will win once. With either bet I have an expected loss of 50 cents. I am again indifferent.
- Likewise I would be indifferent currently between the two bets if I had some probabilistic belief about how I would bet in the other awakening.
So no matter what I believe my other bet will be, I am indifferent between betting on heads or tails currently. This reveals that I believe it is equally likely that the coin has fallen on heads or tails. This despite the fact that, indeed, my payoff is strictly higher if my strategy is to bet on tails at every opportunity. It is a feasible strategy, it is a consistent strategy in the sense that I would have no reason to deviate from it when I actually have to place my bet (point 1 confirms this) and it gives me an expected payoff of 50 cents!
(The truly bizarre thing is that I am indifferent as to how I bet now, but I strictly prefer that the other time I bet tails. The resolution of course is that if there is another time, then indeed tails is the better bet. 🙂 )
(Follow the Wiblin link to see how physicists are (apparently) willing to bet large sums of money on the existence of infinitely many parallel universes, based on a similar example.)