It’s hard to come up with reasonable priors for such theses as Naturalism and Theism and with reasonable conditional probabilities for such evidence as Evils We Can’t Theodicize on Theism. But we can sometimes come up with reasonable comparisons of the strength of evidence. And this might lead to some helpful non-numerical probabilistic reasoning.
For instance, we might have the judgment that the evidential strength of the Problem of Evil (POE) as an argument against theism is no greater than the evidential strength of the Finetuning Argument (FTA) as an argument for theism. Two thoughts in support of this: (1) the low-entropy initial state of the our universe has been estimated by Penrose to be utterly incredibly unlikely (my paraphrase of his 10^(-10^123)) and some of the other anthropic coincidences come with what are intuitively extremely narrow ranges; the theist has proposed various theodicies–they may not be convincing, but it seems reasonable to say that the probability that together they answer the POE is no less, indeed quite a bit greater, than the incredibly tiny probabilities that FTA claims; (2) just as thinking about naturalistic multiverse hypotheses significantly decreases the force of FTA, thinking about theistic multiverse hypotheses significantly decreases the force of POE (cf. Turner and Kraay’s work); (3) just as in the case of FTA we might worry that there is some nomic explanation of the coincidences that we haven’t found, so too in the case of POE we have sceptical theism.
This means that the theist can simply sacrifice FTA to POE: the FTA either balances POE or outbalances POE (I think the latter, because of point (1) above).
Then the theist has a nice supply of other strong and serious theistic arguments, such as the cosmological, non-FTA design arguments (e.g., Swinburne’s laws of nature argument), ontological, religious experience, moral epistemology (theism has a much better explanation than naturalism of how we can know objective moral truths), etc. The atheist has a few other arguments, too, but I think they are not very impressive (the Stone and other issues for the Chisholming of divine attributes, Grim-style worries about omniscience and infinity, worries about the interaction between the physical and nonphysical). At least once POE is completely out of the picture, even if FTA is lost, the theist can make a very strong case.