What is the name of this inference where one uses uniqueness to imply very low probability and design?

Question

Suppose there are 100 trillion humans. Only one of them happens to have three legs and let’s say he is also born in a church on the pope’s birthday.

Suppose a Christian then says, “Perhaps God wanted to send a sign down by making the only person out of 100 trillion have a special quality and also be born in a church during a significant event. The probability of this by chance would be 1/100 trillion.”

Something about this probability seems off instinctually but I’m having a hard time understanding why. Or is it accurate?

Answer 1

If a single three-legged human were born, in a population of 100 trillion, that would be a rare event. But there would be a one in 365 chance that it would be born on the Pope's birthday, so the coincidence being relied upon to attribute divine intervention to the birth is less remarkable by about 12 orders of magnitude. (I do not know how to estimate the chance of a child being born in a church, so I will overlook that.)

Answer 2

The birthday paradox is a surprising probability result. If you have a random group of 23 people, you might think it would be very unusual to find that two of the people have the same birthday. After all, there are only 23 people, but there are 365 possible birthdays. But in fact, there is about a 50% chance that two people in the group will have the same birthday.

This improbable circumstance would fit into that same paradox. After all, it's not like it was general belief ahead of time that if a 3-legged person were born in a church on the Pope's birthday, that this would be miraculous. Rather the miraculousness was inferred after the fact. But it would have been thought just as improbable if a three-armed baby had been born in some saint's home town on the anniversary of the day the saint was sainted.

How many combinations of special place, special day, and special birth circumstance are there? It's open-ended. But give all of the possibilities, the odds that someone will be born on a special day in a special place with some special circumstance are pretty high.

This, incidentally, is how fake prophets make a name for themselves: they make vague predictions that can be interpreted lots of different ways and then look for an event that can be matched to the claim. For example, Q is an internet account that some conspiracy believers think has inside information on government conspiracies. Q posted two comments a couple of days apart, something like (I'm relying on memory here, and my account may be off a bit):

1. Every dog has his day.
2. No-name will be back in the news.

What is he predicting? You can't possibly identify anything. When John McCain died a few months later on the same day as some national dog show, Q fans decided that those two comments had predicted his assassination by agents of the government. But the comments were so open ended that lots of events could have been seen as fulfilling the prediction, and there was a high probability of some coincidence involving dogs and someone who had been out of the news for a while coming back into the news.