Consider this argument:
1. If the PSR is true, then God exists. (Premise)
2. The PSR is false. (Assumption)
3. If the PSR is false, then the probability of any thing's having an explanation is inscrutable. (Premise)
4. The probability of a thing's having an explanation is not inscrutable. (Premise)
5. Hence, (2) is false. (From 3 and 4)
6. Therefore, God exists. (From 1 and 5)
(1) requires some additional argumentation. For example, if the PSR is true, couldn't the sum total of all contingent entities C be explained by its parts, with its parts all being contingent? Well, not if we take seriously the notion that C itself may exist contingently. If C is necessary, then it's necessarily the case that some contingent entity or other exists. This seems highly implausible (for reasons given by Pruss and Craig). It would imply that the non-existence of some entity implies the existence of another. Yet, none of us would say that the non-existence of all non-unicorns implies the existence of a unicorn.
The question still remains: why is there is a sum total of contingent entities at all? Couldn't the whole series have failed to exist? If so, then the standard Humean objection fails to refute the argument from contingency. Given the truth of the PSR, then, it follows that a necessary entity (God) is needed to cause C.