But maybe it's not really a problem at all. The difficulty with the argument is that we have only one universe to refer to. To get any idea about how likely or unlikely are the observed values, we would need a whole raft of universes, each with (possibly) different values of the parameters, so that we could get an idea of what ranges they can lie in and how they are distributed inside those ranges. That is, we need some idea of the probability space that we are working with.
Here's part of a comment I wrote on Luke's site:
Neither here, nor in the Nunley video, nor in the extensive discussion that followed that video do I see anyone address the fundamental issue of fine tuning: namely, that in order to talk about the probability of anything you have to have some idea of the probability space. All the card and firing squad analogies are wildly misleading, because in those cases we KNOW what the probability space is, roughly at least.
It's not like being dealt a royal flush. It's like being dealt a hand in which you don't know how many suits there are, you don't know how many card values there are, and you don't even know what game you're playing. (I could add "you don't even know how many cards you are holding," seeing as we don't know which of the physical constants are independent.)
If you prefer firing squads, it's like you don't know how many marksmen there are, you don't know how far they are standing from you, you don't know how well they are trained, and you don't know how many of them are shooting blanks.
[The comment in question seems to have disappeared from the site, presumably a victim of Luke's recent virus troubles.]