> This one, on the other hand, I don't like. Depending on a whole bunch of subjective definitions, a one-in-billion event can happen a million times a second or practically never or whatever else you choose.
I think this is about events happening to people, the number of people alive (and assuming they all communicate "miracle" occurrences"), and how many things they experience.
That is, if I understand if correctly it's not that you can choose a random number between one and a billion and run a CPU to randomly check numbers in that range as fast as possible and get lots of results in seconds, it's that based one how we have roughly 8 billion people all communicating events that things we consider "one in a billion" occurrences will be experienced about 8 times a month across the populate, and we'll all pretty much hear about it, which may not match with our expectations of how often we should see a "one in a billion" event reported.
Edit: Here's some relevant info from the paper "Methods for Studying Coincidences"[1]:
The Law of Truly Large Numbers. Succinctly put, the law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen. The point is that truly rare events, say events that occur only once in a million [as the mathematician Littlewood (1953) re- quired for an event to be surprising] are bound to be plentiful in a population of 250 million people. If a coin- cidence occurs to one person in a million each day, then we expect 250 occurrences a day and close to 100,000 such occurrences a year.
Going from a year to a lifetime and from the population of the United States to that of the world (5 billion at this writing), we can be absolutely sure that we will see incred- ibly remarkable events. When such events occur, they are often noted and recorded. If they happen to us or someone we know, it is hard to escape that spooky feeling.
A Double Lottery Winner. To illustrate the point, we review a front-page story in the New York Times on a "1 in 17 trillion" long shot, speaking of a woman who won the New Jersey lottery twice. The 1 in 17 trillion number is the correct answer to a not-very-relevant question. If you buy one ticket for exactly two New Jersey state lot- teries, this is the chance both would be winners. (The woman actually purchased multiple tickets repeatedly.)
We have already explored one facet of this problem in discussing the birthday problem. The important question is What is the chance that some person, out of all of the millions and millions of people who buy lottery tickets in the United States, hits a lottery twice in a lifetime? We must remember that many people buy multiple tickets on each of many lotteries.
Stephen Samuels and George McCabe of the Depart- ment of Statistics at Purdue University arrived at some relevant calculations. They called the event "practically a sure thing," calculating that it is better than even odds to have a double winner in seven years someplace in the United States. It is better than 1in 30 that there is a double winner in a four-month period-the time between win- nings of the New Jersey woman.
1: https://www.gwern.net/docs/statistics/bias/1989-diaconis.pdf