One of my favourite mind-bending topics is probability theory. It turns out that, for some reason, human beings are very bad at grasping how probability works. This is evident in many phenomena: why do we think the roulette wheel is more likely to come up black after a long string of reds? Why do people buy lottery tickets? Why is it so freakin’ hard to convince people to switch doors in the famous Monty Hall Dilemma?
Part of the problem is that we seem to think we understand probability much better than we actually do. This is why card sharks and dice players continue to make a living by swindling people who fall into common traps. Studying probability is one of the most humbling things a person can do. One area that has particular relevance to physics is the concept of anthropic reasoning. We base our decisions on prior knowledge that we possess. But it is not always obvious which prior knowledge is relevant to a given problem. There may be some cases where the mere knowledge that you exist — in this time, as yourself – might conceivably tell you something useful.
The anthropic argument in cosmology and physics is the proposal that some observed facts about the universe can be explained simply by the fact that we exist. For example, we might wonder why the cosmological constant is so small. In 1987, Steven Weinberg argued that if it were any bigger, it would not have been possible for life to evolve in the universe — hence, the mere fact that we exist implies that the value of the constant is below a certain limit. However, one has to be extremely careful about invoking such principles, as we will see.
This blog post is likely to be the first among many, in which I meditate on the subtleties of probability. Today, I’d like to look at an old chestnut that goes by many names, but often appears in the form of the `Adam and Eve’ paradox.
Adam finds himself to be the first human being. While he is waiting around for Eve to turn up, he is naturally very bored. He fishes around in his pocket for a coin. Just for a laugh, he decides that if the coin comes up heads, he will refuse to procreate with Eve, thereby dooming the rest of the human race to non-existence (Adam has a sick sense of humour). However, if the coin comes up tails, he will conceive with Eve as planned and start the chain of events leading to the rest of humanity.
Now Adam reasons as follows: `Either the future holds a large number of my future progeny, or it holds nobody else besides myself and Eve. If indeed it holds many humans, then it is vastly more likely that I should have been born as one of them, instead of finding myself rather co-incidentally in the body of the first human. On the other hand, if there are only ever going to be two people, then it is quite reasonable that I should find myself to be the first one of them. Therefore, given that I already find myself in the body of the first human being, the coin is overwhelmingly likely to come up heads when I flip it.’ Is Adam’s reasoning correct? What is probability of the coin coming up heads?
As with many problems of a similar ilk, this one creates confusion by leaving out certain crucial details that are needed in order to calculate the probability. Because of the sneaky phrasing of the problem, however, people often don’t notice that anything is missing – they bring along their own assumptions about what these details ought to be, and are then surprised when someone with different assumptions ends up with a different probability, using just as good a logical argument.
Any well-posed problem has an unambiguous answer. For example, suppose I tell you that there is a bag of 35 marbles, 15 of which are red and the rest blue. This information is now sufficient to state the probability that a marble taken from the bag is red. But suppose I told you the same problem, without specifying the total number of marbles in the bag. So you know that 15 are red, but there could be any number of additional blue marbles. In order to figure out the probability of getting a red marble, you first have to guess how many blue marbles there are, and in this case (assuming the bag can be infinitely large) a guess of 20 is as good as a guess of 20000, but the probability of drawing a red marble is quite different in each case. Basically, two different rational people might come up with completely different answers to the question because they made different guesses, but neither would be any more or less correct than the other person: without additional information, the answer is ambiguous.
In the case of Adam’s coin, the answer depends on things like: how do souls get assigned to bodies? Do you start with one soul for every human who will ever live and then distribute them randomly? If so, then doesn’t this imply that certain facts about the future are pre-determined, such as Adam’s decision whether or not to procreate? We will now see how it is possible to choose two different contexts such that in one case, Adam is correct, and in the other case he is wrong. But just to avoid questions of theological preference, we will rephrase the problem in terms of a more real-world scenario: actors auditioning for a play.
Imagine a large number of actors auditioning for the parts in the Play of Life. Their roles have not yet been assigned. The problem is that the director has not yet decided which version of the play he wishes to run. In one version, he only needs two actors, while in the other version there is a role for every applicant.
In the first version of the play, the lead actor flips a coin and it comes up heads (the coin is a specially designed stage-prop that is weighted to always come up heads). The lead actress then joins the lead actor onstage, and no more characters are required. In the second version of the play, the coin is rigged to come up tails, and immediately afterwards a whole ensemble of characters comes onto the scene, one for every available actor.
The director wishes to make his decision without potentially angering the vast number of actors who might not get a part. Therefore he decides to use an unconventional (and probably illegal) method of auditioning. First, he puts all of the prospective actors to sleep; then he decides by whatever means he pleases which version of the play to run. If it is the first version, he randomly assigns the roles of the two lead characters and has them dressed up in the appropriate costumes. As for all the other actors who didn’t get a part, he has them loaded into taxis and sent home with an apologetic letter. If he decides on the second version of the play, then he assigns all of the roles randomly and has the actors dressed up in the costumes of their characters, ready to go onstage when they wake up.
Now imagine that you are one of the actors, and you are fully aware of the director’s plan, but you do not know which version of the play he is going to run. After being put to sleep, you wake up some time later dressed in the clothing of the lead role, Adam. You stumble on stage for the opening act, involving you flipping a coin. Of course, you know coin is rigged to either land heads or tails depending on which version of the play the director has chosen to run. Now you can ask yourself what the probability is that the coin will land heads, given that you have been assigned the role of Adam. In this case, hopefully you can convince yourself with a bit of thought that your being chosen as Adam does not give you any information about the director’s choice. So guessing that the coin will come up heads is equally justified as guessing that it will come up tails.
Let us now imagine a slight variation in the process. Suppose that, just before putting everyone to sleep, the director takes you aside and confides in you that he thinks you would make an excellent Adam. He likes you so much, in fact, that he has specially pre-assigned you the role of Adam in the case that he runs the two-person version of the play. However, he feels that in the many-character version of the play it would be too unfair not to give one of the other actors a chance at the lead, so in that case he intends to cast the role randomly as usual.
Given this extra information, you should now be much less surprised at waking up to find yourself in Adam’s costume. Indeed, your lack of surprise is due to the fact that your waking up in this role is a strong indication that the director went with his first choice – to run the two-person version of the play. You can therefore predict with confidence that your coin is rigged to land heads, and that the other actors are most probably safely on their way home with apologetic notes in their jacket pockets.
What is the moral of this story? Be suspicious of any hypothetical scenario whose answer depends on mysterious unstated assumptions about how souls are assigned to bodies, whether the universe is deterministic, etc. Different choices of the process by which you find yourself in one situation or another will affect the extent to which your own existence informs your assignation of probabilities. Specifying these details means asking the question: what process determines the state of existence in which I find myself? If you want to reason about counterfactual scenarios in which you might have been someone else, or not existed at all, then you must first specify a clear model of how such states of existence come about. Without that information, you cannot reliably invoke your own existence as an aid to calculating probabilities.