Probabilities and Probability Densities
Although one may not doubt that Penrose's figure of 1:10^10^123 is a good stab at the relative volumes in phase space (that is, the collection of all possible universes), this value, in and of itself does not give us theprobability that, given a random selection of points in phase space, ours would be chosen. Nor has it ever been demonstrated that the origin of the universe would include some sort of random processes whereby the values of the fundamental constants were determined.
Why should we not treat the Penrose figure as describing the improbability of our universe? Because any calculation of any probability requires a knowledge of the relevant probability densities. So, the calculation of our particular universe existing would require expressions for the probability densities for various universe scenarios. And I think it is safe to say thatnobody knows what the probability densities are at this time, or even if the concept of "various universe scenarios" is at all meaningful.
To get a quick feel for the meaning and necessity of probability densities, consider this simple example, which involves a discrete random variable: Suppose you roll two fair dice simultaneously. The "phase space" consists of the numbers two through twelve. The actual value you roll will be just one value within that phase space, one of eleven possible values. However, this does not mean that all of the outcomes of a random toss are equally likely. For example, you are more likely to roll a six than a two, because there are more ways to obtain a six than a two. In other words, the outcome for the two dice has a non-uniform probability density, because some outcomes are more likely than others.
Another illustration demonstrates the same idea, but with a continuousprobability density. Suppose you have a clock with only a minute hand, and that the minute hand has become disconnected from the inner mechanism and is now free to spin about. If the clock is lying horizontally and you give the hand a spin, it will go around a number of times and then settle on some number. What are the odds of any random spin resulting in the hand, or pointer, pointing, say, between the three and the four? Since there are twelve such intervals where it may stop, the probability, assuming that it spins smoothly, is 1/12. Here the probability density is uniform. The pointer is just as likely to point to any given spot as any other after a random spin. Now, consider what happens if you hang the clock back on the wall. What is the probability of obtaining a value between three and four now? Has the probability density changed? Yes, if you spin the pointer, it will now point to the six (straight down) with 100% (or close) probability. The probability density has been drastically altered, even though we have not changed the set of all possible outcomes and the phase space.
How does this relate to the problem at hand? How could we determine the probability densities for all possible universes? The direct, empirical method would involve looking at the statistics of a large number of universes. One could estimate the densities by examining the properties of other, diverse universes picked at random. Unfortunately, that method is clearly out of the question. We only have one universe to look at. Unless there is a theoretical model which predicts these probability densities (along with other results which can be tested), we have no reason to conclude they are uniform (as so many have done by default) or bell-curves or whatever. (Efforts in the field of quantum gravity and string theory may eventually yield the values of the fundamental constants as part of a theoretical framework, but this has yet to be seen.) And this means one should be wary of using the term "fine-tuned" to describe the values of the constants (just as one would be wary of the answer to a calculus problem when no limits of integration and no integrand were specified). If the probability densities for the fundamental constants were Dirac delta functions,[9] for example, then our universe would have come into existence with a probability of unity and the concept of tuning would clearly not even apply.
Or, to modify the biased analogy of the AD proponents, we don't know if a firing squad of 100 marksmen or a single blind man with a musket is a better representation of the conditions surrounding the early moments of the universe.
So, the proper response to that favorite rhetorical question of the AD proponents, "Come on, what are the odds that those constants were tuned by chance?" is: "Yes! Indeed! What are the odds? What are the probability densities?"