There is a subset of the supernatural being fan club whose members are enamored of improbability arguments. That is, they will calculate the probability of some feature of the universe, get a very small number, and declare that since the probability is so low under a naturalistic scenario, supernatural intervention is required.
Usually there are some blatant surface errors in such a calculation, such as assumptions no one is entitled to, or an artificially narrow set of naturalistic scenarios. But there are also deeper concerns. “Inverse problems” are notoriously nasty, and inferences to supernatural causes based on alleged improbabilities is a particularly good place to find pathologies.
Try these examples out for size:
1 (Strong version): The probability of the universe is exactly zero. If lower probabilities mean increased credibilty for a supernatural explanation, then we can conclude with certainty that God created the universe.
As a matter of fact, the probability of the exact state of our universe is (probably) zero. That is, it seems reasonable to say that there are an uncountable infinity of possible states of the universe. In that case, whatever state we occupy is trivially a zero measure set. Its probability is exactly (not approximately) equal to zero. Short of logical impossibilities, it’s hard to find a better candidate for divine causation on account of improbability.
2 (Weak version): As we increase our knowledge about the universe, we automatically decrease the probability of the universe. Therefore, as science advances, the likelihood that there is a God inevitably increases.
Our universe is constrained by the (finite) information we have about it. As we increase our information, we eliminate possible universes that do not conform to the new information from the set of possibilities. Therefore the probability of the information we have decreases as the amount of information increases.
I hope the bullshit nature of these arguments is obvious. Many supernaturalists think they can infer a supernatural cause based only on improbability of a data set, without specifying any new pattern in the data that is explained by a supernatural agent. You simply can’t do that—it’s merely a failure to recognize randomness. The pathology here is not merely in using inappropriate statistical models and so forth, though often these show up as devices to inflate alleged improbabilities. (The above arguments are independent of such concerns.) Even if you get the probabilities right, you are not entitled to infer supernatural agency. Improbability is not a excuse to get out of doing real, novel explanatory work.
But then again, I do think that a major thread through almost all intellectualized supernatural belief is a failure to appreciate randomness.