Off-site someone linked me to this post by Richard Carrier, in which he comes to much the same conclusions as I did, but in much more depth, and also provides a more formal logical proof.

P1:In the beginning, there was absolutely nothing.P2:If there was absolutely nothing, then (apart from logical necessity) nothing existed to prevent anything from happening or to make any one thing happening more likely than any other thing.C1:Therefore, in the beginning, nothing existed to prevent anything from happening or to make any one thing happening more likely than any other thing.P3:Of all the logically possible things that can happen when nothing exists to prevent them from happening, continuing to be nothing is one thing, one universe popping into existence is another thing, two universes popping into existence is yet another thing, and so on all the way to infinitely many universes popping into existence, and likewise for every cardinality of infinity, and every configuration of universes.C2:Therefore*, continuing to be nothing was no more likely than one universe popping into existence, which was no more likely than two universes popping into existence, which was no more likely than infinitely many universes popping into existence, which was no more likely than any other particular number or cardinality of universes popping into existence.P4:If each outcome (0 universes, 1 universe, 2 universes, etc. all the way toaleph-0 universes, aleph-1 universes, etc. [note that there is more than one infinity in this sequence]) is no more likely than the next, then the probability of any finite number of universes (including zero universes)or lesshaving popped into existence is infinitely close to zero, and the probability of some infinite number of universes having popped into existence is infinitely close to one hundred percent.C3:Therefore, the probability of some infinite number of universes having popped into existence is infinitely close to one hundred percent.P5:If there are infinitely many universes, and our universe has a nonzero probability of existing (as by existing it proves it does, viacogito ergo sum), then the probability that our universe would exist is infinitely close to one hundred percent (because any nonzero probability approaches one hundred percent as the number of selections approaches infinity, via the law of large numbers).C4:Therefore, if in the beginning there was absolutely nothing, then the probability that our universe would exist is infinitely close to one hundred percent.