I would say that theists have a hard time dealing with randomness, but that would be misleading. Quite a few nonbelievers also dislike randomness. Randomness offends against the intuition that everything has a cause, whether this eventually means supernatural ultimate causes or a universe where every event has a natural cause.
Still, religious thinkers seem to have a particular animus against randomness, continually speaking about the absurdity of it all coming down to chance. Those parts of modern science that present random aspects of nature attract religious suspicion. Quantum mechanics is full of randomness. Darwinian evolution relies on random variation to supply the novelties selection then works upon. They create theological problems.
Creationists solve their problem by denying and denouncing randomness. But randomness, by suggesting lack of design and direction, challenges more science-friendly theists as well.
Divine Action and Natural Selection has numerous examples of what is one of the standard liberal religious responses to randomness: saying that apparent randomness is compatible with a God running the show. After all, it is possible that if we take a broader view than just physics or biology, what we thought was random might just be God’s way of accomplishing divine purposes. Rabbi Natan Slifkin, for example, points out that
One point to note is that it is impossible to ever determine whether something is truly random. One could take a string of a hundred seemingly random numbers and perform every conceivable test, and discover no pattern. Yet it could be that those numbers were actually the numbers preceding the one hundred digits that appear after the millionth digit of Pi. So, they were not actually random at all. [p. 623]
Indeed, this is absolutely correct. Randomness is, strictly speaking, a property of infinite sequences. By looking at a patternless (algorithmically random) finite sequence, we can guess that the finite data is part of a random sequence. But this inference is always fallible.
But that misses the point! Mere compatibility between science and God is very cheap and easy to obtain. A creationist who declares that God placed fossils in the earth to test the faith of believers is also achieving compatibility, in much the same way. The real question is what is the best model, the best explanation to account for the finite but patternless data. In that context, bare compatibility is not all that relevant. It’s still possible that what we thought was random was only apparent randomness, but we cannot infer this on the basis of the patternless data. We need further information. It may be that an intelligent designer is responsible for quantum fluctuations or the contingencies of evolution. But we cannot infer this from present-day physics or biology. We need some other reason to think there is a God behind the scenes, pulling the strings of what is apparently random. For example, a clear and convincing revelation expressed in some kind of scripture might work. Good luck with that. In the absence of a pattern revealed by bringing in a wider range of data, scientists are perfectly correct to say that they’re dealing with actual randomness.
I don’t want to pick on Slifkin; he’s just the easiest one to quote. There are others in Divine Action and Natural Selection who make the same error, even when they should know better. For example, Stephen M. Barr, who is a legitimate and respectable particle physicist, but who is also a conservative Catholic with ambiguous intelligent design leanings. Indeed, Barr has become a favorite among some conservative Catholic intellectuals in the US, such as Michael Novak.
A shorter exposition of Barr’s thoughts on randomness and evolution can be found in his First Things article, “The Design of Evolution.” After Barr’s paper in Divine Action and Natural Selection, you can find my critique, as part of a dialogue between us. But in short, it’s the same mistake, made worse by errors in Barr’s technical understanding of randomness. (Why does a physicist, when reflecting on randomness, not refer to the relevant mathematical literature, rather than drowning the concept in theological gobbledygook?)
I’ll stop here before this turns into even more of a rant. In any case, randomness is one of my personal hobbyhorses, and one of the major themes in my books. If you’re interested, read them to find out more.