An Objection to Simplicity Style Arguments in Favor of Theism

The Objection: While the hypothesis of an unlimited foundation might be more probable than the hypothesis of a foundation with some specific limit, that ignores the following: there are infinitely many ways to be limited *in some way or another*, and only one way to be unlimited. And so even if being unlimited is more probable than any given limit chosen at random, surely 'the foundation is limited in some way or another' is far, far more probable than 'the foundation is utterly unlimited.'

Reminds me of the man who shoots himself in the head:  There’s an infinite number of highly unlikely scenarios where he doesn’t die.  Perhaps it misses his brain, perhaps the powder is wet, perhaps the trigger fails, perhaps he twitches, and so on.  And yet the infinite or near infinite scenarios where he happens to survive doesn’t swamp the most likely scenario - they shouldn’t lead us to think that he’s not almost certainly going to die.

There’s probably a more principled approach that avoids this objection. Here's an attempt: The mere existence of infinitely many options doesn't make them all equally probable with one another. There are infinitely many alternatives to general relativity, but they are probably all false and general relativity is probably true.

But how is it that an infinite number of alternatives, even if all singularly very unlikely, do not swamp the probability of the most likely single explanation? The probabilities of the infinitely many alternatives could form a converging series. Consider the series:

1/2, 1/4, 1/8, 1/16, ...

This has infinitely many terms. But their sum is not infinite: it is just 1. Now if those are probabilities expressed as percentages, the first one is 0.5%, the second is 0.25%, the third is 0.125%, etc. Then the sum of the (infinitely many!) probabilities, which is the probability that one or another of them is true, is just 1%. Not too impressive.

From Swinburne: Consideration of the weight we give to simplicity in other areas of inductive inquiry suggests that we normally give it such weight that a really simple hypothesis is intrinsically more probable than a disjunction of many more complex hypotheses.  For example-would a detective really think it is more probable that some hypothesis or other to the effect that Jones cooperated with at least one of the few billion inhabitants of the earth to do some crime more probable than the hypothesis that Jones did it alone when both hypotheses give equal probability to the observed data?