In some corner of the multiverse, there exists a universe-bubble with physics close enough to our own to be recognizable, but different enough to make colonizing the stars feasible. In some point in this universes history, there was a Galactic Empire that ruled over many zillions of humans with an iron fist, imposing its religion upon all its subjects.
According to the theology of the One True Faith, the gods were just. The gods cared deeply that judicial procedures and punishments be carried out painstakingly. The Empire committed enormous resources to its judicial system, run, naturally, by the priesthood. The worst crimestreason, heresy, blasphemy (much the same sort of thing, according to the priests)deserved a particularly nasty capital punishment. The condemned person was to be put in front of a special pulsed laser cannon produced at enormous expense, whose components were made of sacred diamonds, and which produced a pulse sequence coding for all the details of the most sacred hymns in the Holy Book of Retribution. And the condemned was to be shot in the middle of the forehead, dying with Truth branded on her brain.
The gods of the One True Faith were also merciful. While demanding the highest standards of their trials and sacred execution devices, they also knew that humans were not perfect. So the Law allowed that if the beam from a laser cannon was to miss the condemned, this would mean that the gods had caught and corrected a mistake. The priesthood was commanded to make their instruments as reliable as possible, but also enjoined to allow for divine intervention.
The diamantine laser had to be prepared with the best available technology, according to precise ritual specifications. The best scientists in the Empire came together to calculate the probability of the laser not hitting the center of the condemned heretic’s forehead. They found that there was only a one in a zillion chance of the beam not hitting the precise correct spot, but still causing death. This would be a sign that the heretic was guilty, but of some sin other than that determined by the Imperial Courts. And there was also a one in a zillion chance that the beam would miss the condemned, leaving her alive. Then she was to be set free.
These probabilities of an off-center beam could not be determined by experiment, by shooting the laser a few hundred zillion times and observing the outcomes, as this would be way too expensive, even for the Empire. In fact, there were even priests who thought it might be sacrilege to shoot a pulse coding for the Book of Retribution at a non-live target. But many of the components could be empirically tested, and the Imperial College of Science had excellent methods to calculate failure probabilities for a complex assembly of components. Indeed, the very success of the Star Drives that led to the conquests forming the Empire depended on the very same methods, and the Star Drives were extremely reliable.
The theologians of the Empire argued for letting the condemned in a failed execution go by the same impeccable reasoning to Intelligent Design that they used to establish the existence and guidance of the gods. After all, an event that was very improbable according to physics would have taken place. And this was not just any improbability: it fell in a very meaningful, pre-specified set of possibilities. Someone, very improbably, ended up alive. Clearly this could not happen just by accident. It had to be a complex specified event that was therefore intelligently designed.
The oblivious physicist was summoned to the capital planet for the occasion, and given charge of the execution device, which he immediately recognized as an unnecessarily elaborate pulsed laser. He triggered it, and soon experienced great outcries of surprise and prayer. The beam had grazed the accused arch-heretic, badly mutilating her in the cheek but still allowing her to live. She was therefore let go, and allowed to live out her life in a monastic planet under a vow of silence.
Soon after the missed shot, however, the physicist started asking what really happened. He went through painstaking checks of the diamantine laser, recording everything, sending information out to experts and also doing his own calculations. He eventually confirmed that absolutely nothing looked tampered with, and that the one in a zillion probability calculations were perfectly correct. All the evidence showed that the laser was operating normally.
Now, the physicist wondered again, what happened? There could have been a mistake, or perhaps even some worldly conspiracy that tampered with the laser in order to save the heretic and then covered its tracks. But that seemed much more implausible than a one-in-a-zillion chance allowed by the physics. So, if dumb luck or some sort of intelligent cause that transcends physics were the only two real options in play, which one was the better bet?
And here, to make it more interesting, are some other questions.
First, assume that the physicist was so oblivious about events that he had no idea how many times the laser had been used in previous executions. For all he knows, it could have been the first and only time. How should he reason in such circumstances?
Second, say this was the one-hundredth execution, and the physicist knows this. The other ninety nine resulted in the heretic being fried straight in the forehead. Does knowing this change anything?
Third, assume that half the galaxy had recently been convulsed by religious wars, and that Imperial Troops resorted to mass killing, ritually exterminating zillions of captured heretics using identical diamantine lasers. The expected rate of a handful of survived executions occurred during these exterminations. The physicist knows this. How would his reasoning change?