What Are the Chances?

Tony Pasquarello

Actually, the night had been neither stormy nor dark. But her sleep was restless and fitful; virtually nonexistent. Temperature was a big part of the problem: she must have added to, removed, and adjusted the bedding a half-dozen times. And now, the wee hours—5 a.m., 6 a.m. When the phone rang, she was definitely in a hypnopompic state, foggy and groggy. But, in just a few moments, she’d be hyper-alert. The voice went on and on, with each wrenching sentence adding bitter details of the tragedy; she felt the predictable effects in her eyes, her throat, her whole being. Suddenly, she gasped. She saw … no … she was transfixed by the digital clock. Its crimson numerals—searing, burning digits—branding her brain with their uncanny significance. She realized the shocking truth: the call had come at the precise time showing the birthdays of the couple involved! That couldn’t be just a coincidence. No way! What are the chances? They must be one-in-a-million. Or maybe a billion.

Where does this sort of event fit in the usual paranormal categories? One website lists more than thirty species of the paranormal. Everyone recognizes telepathy, clairvoyance, psychokinesis, and remote viewing—along with ghosts and spirits.

Devising a taxonomy of the paranormal can be frustrating, because we never get to touch base with reality; we never get to examine a genuine specimen from the vast, untidy, paranormal zoo. The task is something like herding imaginary cats.


Here is another event that would qualify as “more than coincidence”: In an accident 2,000 miles from home, the person saving your life has your identical first and last names. However, if your hotel clerk had your name, that probably would seem “just a coincidence.”

I was prepared to claim that these sorts of phenomena have not been given sufficient consideration nor any handy generic term. Then, just a bit more internet exploration revealed otherwise. No less a luminary than Carl Jung was discussing “meaningful coincidences” just about a hundred years ago and developed his ideas more fully at mid-century. He referred to these events as “synchronicities” and saw them as “expressions of a deeper order.” Apparently, the interconnectedness of the collective unconscious occasionally bubbles up into the public, conscious sphere.

However, neither the term synchronicities nor the definition as “temporally coincident occurrences of acausal events” captures the very thing that makes them interesting and controversial—their surprising nature, the assumption of extreme improbability that rules out chance as the explanation. After all, for any given moment, there are an infinity of synchronous, acausal events, viz. almost all the pairs occurring at that time. But the events in question are those for which it is claimed that they cannot be “mere” coincidence—the claim is that something more is involved, an element that “paranormalizes” the coincidence.

At least one thing is fairly clear—the term synchronicities never caught on. I instead suggest Coincidence Plus (C +) as a more convenient class term, where the “+” represents the allegation that something more than coincidence is involved. C+ also handily covers concurrent, convergent, and all the other “con” terms that might be appropriate. And for a retired professor, inscribing “C+” is a familiar activity.

Muddled Millions

C+ events are those that, almost reflexively, elicit the rhetorical exclamations (pseudo-judgments) “one in a million” or “a million to one”—the perennial, presumed answers to the rhetorical title question, “What Are the Chances?” Actually, references to a million as some sort of zenith stem from the first half of the twentieth century. The “million-dollar baby from a five & ten-cent store (selling china)” provides a glimpse into the economy a century ago. I recall wondering, in the 1940s, whether my dad could ever reach the ultimate middle-class pinnacle of $10,000 a year. Those were the days when Rockefeller was just about the only millionaire who came readily to mind. But a “million to one” was usually the wrong answer then and is still the wrong answer now that the U.S. population has tripled.

Recognizing that various “million” slogans are rarely serious probability estimates, nevertheless, constant repetition has a way of creating a psychologically favorable climate. In the current political ethos, it is often noted that repeating a lie often enough can make it seem true. Repetition legitimizes the subject and suggests possibility, if not actuality. Natural humility in skeptics, perhaps most people, makes them reluctant to call anything impossible. Just so can the rhetorical be taken as literal; repetition creates reality. Just consider, for example, the freakish file of bizarre paranormals stalking across our TV screens every evening. Even a commonsensical person could be forgiven for asking “Can they all be imaginary?” By the way, the correct answer is “Yes!”

Even the “million-dollar baby”—if the implication is that her charm is one in a million—is grossly misleading. In fact, the ratio of very attractive women—or men—is closer to one in fifty. Indeed, if some event that applies to all really has an empirical probability of a million-to-one on any given day, then, by definition, that “improbable” happening is happening about 330 times in the United States today! If you live in a fairly large city, it’s happening near you two or three times! Or, consider the chances of winning $10,000: seems quite remote, right? Then think of all the casinos, lotteries, drawings, and of course Las Vegas. It’s happening, frequently—and it happened to me!

Another facet of the problem was recently brought home by a striking coincidence—but not a C+. While working on this very article, I overheard some TV character saying “There were a million or a billion.” Naturally, my ears perked up. Upon reflection, I decided that I had heard a notably crazy disjunction: the speaker was uncertain about something being a certain size—or instead one thousand times larger! That would be like being unsure whether a gathering was the size of the Buckeye football team or the capacity crowd at Ohio Stadium. And that is obviously absurd. People don’t usually make quantitative mistakes on the order of a thousand-fold! I suspect that the script writer was prone to a common error—thinking that billion means “two million.”

Getting Down to Cases

In crafting a sensible response to the title question, we have to abandon the murky swamp of the paranormal and the amorphous clouds of the hypothetical and do the obvious—a real world analysis. I decided to select an average, four-person, Midwest, middle-class family—mine. It consists of two children, a boy and a girl, two years apart. These are the actual birthdates:

  1. 4-13-33
  2. 8-27-33
  3. 8-17-58
  4. 8-07-60

A considers 3 and 4 to be special, significant numbers, while the other three favor 7 and 8.

It is essential to directly address a possible charge that my “average” family is a biased sample because of the latter three covering all the sevens in August. While that may appear to be slightly unusual, think of the thousands of four-person families where all four members evince some pattern or other. Then, think of the virtually limitless number of “orderly” arrangements that the numbers 0–60 can exhibit for three or four persons. These range from simple serial succession (22, 23, 24, 25) to arithmetic sequences (2, 4, 8, 16; 3, 6, 9, 12) to more esoteric sequences such as pi, e, or Fibonacci. The number will be very large indeed.

We are trying to provide a concrete answer to the following question: What are the chances that a phone call disclosing some significant (even traumatic) event will come at the same time as the digital clock’s showing the significant dates (birthdays, etc.) of the event participants? The call may come from one of the family separated from the others or from another person. If the latter, more numbers will become available targets.

A reasonable assumption is that such a call, especially if the event occurred in the very early morning hours, would come between 7 a.m. and 8 a.m., out of consideration for the person called.

If we have all the relevant data and have correctly partitioned the complex event into appropriate subsets, all is ready. After all, in calculating probability, it is not that the math is hard; knowing how to analyze the situation to decide which items will be going into the calculations can be the difficult part.

From 7 a.m. to 8 a.m., the clock will show sixty different times. Given the foregoing, these will be, prima facie, “significant” hits: 7:00, 7:03, 7:04, 7:07, 7:08, 7:13, 7:17, 7:27, 7:33, 7:37, 7:47, 7:58.

I’d plead the case for two others as obviously significant: 7:11, a commonly understood symbol for speedy shopping—particularly, off-hours. For any activity remotely connected to shopping, 7:11 will be taken as a hit. Then, 7:31, easily seen as reversing the bad luck of 13, can be a hit for any event involving “reversal”—an accident while backing up, exchanging usual lunch seating, etc.

The clear, first-stage answer to the title question/scenario is fourteen out of sixty, very nearly 25 percent—the probability of this type of C+ event. Needless to say, nothing like a million-to-one.

Moderate Manipulation

Suppose that the event in one case is somehow related to an unusual meeting between B and D. Then, if the idea occurs to someone, the time 7:34 (27+7) could be taken as a C+ hit. Likewise, for 7:20 (27-7) if the two are separated. The time 7:15 (15=604) might be a hit if a heated political argument divides A and D. Permitting just one arithmetic or quasi-arithmetic step produces a host of new possibilities.

The four-person table contains twelve one- and two-digit numbers, specifying birthdays. Taken two at a time, there are sixty-six possible pairs. The numbers in each pair can be added or multiplied (producing two new numbers); subtracted or divided (four new numbers); or exponentiated (two new numbers). They can also be conjoined, placed alongside each other—a nonmathematical operation—in two different ways. Hence, simple one-step operations produce a lot of new candidates for C+ status, even though many will be too large—e.g., —to be relevant. So the probability of a C+ increases.

Recognizing that the family’s special numbers—3, 4, 7, and 8—constitute direct hits, what of attempting to evade those numbers? What if the call comes at 2:59, 5:29, or 9:52? Well … 2+5=7; 5-2=3; 9-5=4; etc. Simple steps and there are the special numbers. How about 10:12? It’s too easy: 1+1+2=4; 10+1+2=13; etc.

Given almost any small group of small numbers, some juggling—just a few simple steps, with a plausible justification for those steps—will produce the desired number or pattern. We might call this property of numbers their “pliability” or “patternability.”

Ambiguity and Vagueness

Often, in actual usage, ambiguity and vagueness are considered negative characteristics; they’re bad things about language to be eliminated. But for our purposes (context is always the arbiter), it is just a fact that most terms have more than one meaning or large gray areas of applicability.

There is nothing remotely ambiguous or vague about the numbers themselves, or the mathematical operations, whatever the ontological status of numbers may be. But the symbols standing for those numbers and operations do exhibit the normal, generous helpings of ambiguity and vagueness.

Consider some cardinal number homonyms: one-won; two-to-too; four-for; eight-ate; nine-nein. The ordinals—second, fourth, and fifth—are obviously ambiguous, with strong connotations. Number terms in other languages could come into play, as could Roman numerals.

When we turn to the operations, we find a host of synonyms, many with strong emotive associations. Addition: plus, join, combine, meet. Subtraction: minus, lessen, take away, remove. Multiplication: procreate, proliferate, times. Division: separate, into, take apart. Exponentiation: raised, to the power.

Suppose the “call” comes at 2:25, 2:52, or 5:22 and involves a “7” person and some achievement related to dance. For example, someone is finally named prima ballerina after numerous attempts. The time can then be interpreted as uncannily predictive: 2+5=7; 5=Roman Numeral V=V for victory; 22=two-two=tutu. Voila! A classic C+ event.

Absurd! Outrageous! Of course—but not to the fantasy prone, paranormally inclined mentality, anxious to find the hand of the supernatural in every possible unusual occurrence.

Furthermore, coincidences such as our initial example constitute a fertile field for the application of a type of retrofitting—not necessarily devious—so often discussed by Joe Nickell. The retrofitting actually creates the C+. Consider how loose the concept of the “call” is. The first ring was really at 7:15; phone answered at 7:16; event announced at 7:17; call ends at 7:21. Any or all of these could be termed “the call.” A recipient of the call who had not noted the clock could be thinking of the dramatic news sometime later, still either distraught or exhilarated. If the event involved persons D (7) and C (17), it would be quite understandable if retrofitting occurred—“My God, the call must have come at 7:17!” (Even if it came at 7:49!) Thus a C+ event that never happened is created without deliberate guile.

Same Birthdate Paradox

For the thirty or so folks at the bar, it is a very good bet (70 percent) that at least two will have the same birthdate (month and date). Most people hearing this are astonished, even disbelieving. Why? Explaining sheds light on the C+ phenomenon, because they are structurally identical. In this case, as in most others, the average person does not know the correct odds and does not know how to go about calculating them. The question is misunderstood as “How many people are necessary for there to be a good chance of matching my birthday?’ Naturally, the mind immediately jumps to 365, the days in a year. Even when the question is clarified, we may persist in thinking the odds are 365 to one, a tiny probability. Psychologically, there is an almost natural reversion to the “one-in-a-million” syndrome, the instinctive reaction to anything perceived as having extremely low probability.

Whether the coincidence is about birthdates, phone calls and clock times, or others, there appear to be two conditions for a C+ event:

  1. The probability of the event is initially thought to be extremely low, such that coincidence can’t be the “whole story”; there must be “something else” at work. (The religious version would often term these events miraculous.)
  2. The event cannot be considered trivial; it must be sufficiently dramatic to hold high emotive significance for those involved.

And the Chances Are …

It is time to answer the title question. Given the stipulated parameters, a C+ event like the “call” has a probability of about 25 percent, a not insignificant figure. That calculation matched only birthdays to clock times. But surely in real life other significant dates, such as ages and anniversaries, will be utilized. If two parties are ages seventy-five and fifty, no doubt 7:50 would be judged a direct, meaningful hit. With four numbers representing ages and three numbers for an anniversary, in many cases, there will be up to seven new numbers for matching. The new ratio could then be 21 out of 60 for a hit. That’s 35 percent.

Permitting just one arithmetic or quasi-arithmetic manipulation surely raises the probability to much better than 50 percent. Finally, a clever zealot with enough creative imagination to finagle the numbers using the additional interpretive possibilities extensively discussed—in that event—a hit is a virtual certainty, no matter what time the call comes.

We have attempted to analyze some of the most prominent and compelling of the C+ events and shown that the probability is almost always much higher than supposed. Hence, in showing that the surprise is logically unwarranted (but understandable), we eliminate the need to invoke the paranormal.

This same psychologically fascinating mindset takes place in a variety of cases. We may initially experience astonishment (wondrous delight or revulsion). Then, in the course of time, we learn the true inner workings of some phenomenon—yet we are still astonished. The moon, flowers, and childbirth (and eating insects, for the revulsion part) come readily to mind. Perhaps the most illuminating analogy can be found in the attitude of the thousands of observers who, this very day, are enjoying the feats of a magician. Both the viewers and the magician know that no real magic is involved; nothing paranormal required. Yet viewers are still mystified and—the magician hopes—enchanted.

A Postscript Probability Puzzle

In that vast aforementioned Ohio Stadium (cap. 105,000), what are the chances that the person seated next to you has the same birthdate?

Were this found to be the case, those involved would probably consider it a C+ event and likely use typical locutions to characterize it: It must be a sign; We were destined to meet; There is an ultimate purpose; This was not by accident; etc.

In solving the puzzle, pay careful attention to initial responses. Did the large number create an adverse reaction? Did the “million-to-one” syndrome rear its ugly head? Did some ambiguity infect your understanding of the problem? Two helpful hints:

  1. About 300 attendees will share your birthdate.
  2. In most cases, there are two people next to you.

In keeping with our thesis, while first impressions may lead someone to assume extremely low probability, the actual chances are not that low. They are nowhere near a million-to-one—and about ten million times better than winning Publishers Clearing House sweepstakes.

Actually, the night had been neither stormy nor dark. But her sleep was restless and fitful; virtually nonexistent. Temperature was a big part of the problem: she must have added to, removed, and adjusted the bedding a half-dozen times. And now, the wee hours—5 a.m., 6 a.m. When the phone rang, she was definitely in …

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