Coincidents (miracles) happen much more often then people expect, all due to randomness.
Description and excerpt from the excellent book “The Canon” by Natalie Angier, Pp 47 – 52. These pages give much more detail and many examples of real life situations.
Also the author gives explanations for the examples.
Teaching elementary statistics students, Deborah Nolan divides her class into 2 groups.
One group is told to take a coin and flip it 100 times and record each flip, Heads or Tails.
The other students are to imagine flipping a coin 100 times and record what the flip should be Heads or Tails. Each student signs their work with a unique mark and places it face down on Nolan’s desk. Nolan leaves the room during this time..
When Nolan returns she looks at all 100 Hs and Ts on each sheet and says which is either a real tossup or imaginary one. Nolan astounds her students by being correct pretty well every time. Nolan knows what real randomness looks like, and she knows that it often makes people uncomfortable by not looking random enough.
“In the real tossing of a coin, flick after flick, you will find many stretches of monotony, strings of five Heads or seven Tails in a row.
In their fantasy flippings, the students compensated for their inherent chariness of “too much coincidence” by frequent hopping back and forth, head to tail. In general, the act of jotting down a triplet would set off an alarm bell in the student’s head, resulting in a deliberate change of face. “When I look at the fabricated coin tosses, the length of the longest run of heads or tails is way too short,” said Nolan. “And overall, the number of switchbacks between heads and tails is way too high.” People know there’s a fifty-fifty chance for a given outcome with each toss, and they know that, on average, one hundred tosses will yield something close to fifty heads and fifty tails. OK, forty-eight tails, fifty-two heads, I can live with that. But six tails in a row?
The author tried Nolan’s coin-tossing exercise herself several times, and over a dozen rounds of one hundred flips each, she never completed a set of one hundred without getting at least one string of six or seven heads or tails in a row, often more than one unbroken sextuplet per set, as well as many quintuplets and quartets. My record for monotony was nine heads.
Here we find the basis for superstitiousness, she said. A chance occurrence occurs. Not knowing the odds behind it, we marvel, Now, really, what are the odds? Surely too tiny for chance! “
I have included 2 examples of many from the book:
John Littlewood, a renowned mathematician at the University of Cambridge, formalized the apparent intrusion of the supernatural into ordinary life as a kind of natural law, which he called “Littlewood’s Law of Miracles.” He defined a “miracle” as many people might: a one-in-a-million event to which we accord real significance when it occurs. By his law, such “miracles” arise in anyone’s life at an average of once a month. Here’s how Littlewood explained it: You are out and about and barraged by the world for some eight hours a day. You see and hear things happening at a rate of maybe one per second, amounting to 30,000 or so “events” a day, or a million per month. The vast majority of events you barely notice, but every so often, from the great stream of happenings, you are treated to a marvel: the pianist at the bar starts playing a song you’d just been thinking of, or you pass the window of a pawnshop and see the heirloom ring that had been stolen from your apartment eighteen months ago. Yes, life is full of miracles, minor, major, middling C. It’s called “not being in a persistent vegetative state” and “having a life span longer than a click beetle’s.”
The more one knows about probabilities, the less amazing the most woo-woo coincidences become. My mother told me an amusing story about an acquaintance of hers whose fate, over a six-month period, had seemed linked to her own as though by an idle Pan. The acquaintance was, appropriately for our purposes, an old math professor of hers. Week after week, my parents kept running into him somewhere on Manhattan’s sprawling cultural turnpike—an off-Broadway play, a free piano recital, a Bergman movie, the Monet Water Lilies room at the Museum of Modern Art. The first few times, my mother and her professor chortled awkwardly over the similarities of their taste. Soon, they were content to nod vaguely from across the room. The coup de graceless came a few months later, in July, and in another country. My parents were strolling along the boulevard St.-Michel on their first trip to Paris, when who should they see but the good professor, sitting at a café. Judging by the way he held his newspaper ostentatiously in front of his face, my mother knew he had spotted them first.
Naturalism and Religious Experience 