The Paradoxical Nature of Coincidence

I am no mystic. By nature, I believe that I am inclined towards scepticism. Occasionally, however, events have occurred that appear to defy reason or scientific explanation. Coincidence is a good example. The traditional scientific explanation holds that coincidences are inevitable within the laws of chance. An alternative explanation entertains the possibility of an as-yet undefined, ‘paranormal’ mechanism existing beyond known science.  

I describe here a coincidence with statistical odds of one quintillion to one (10-18). This example poses a dilemma because there is no logical way of confirming or disconfirming either of the two theories. The usual procedures for testing statistical significance cannot be applied. Whether this particular coincidence is a chance event or evidence of the paranormal must remain a subjective judgement.

Two Opposing Theories

My intention is to explore two contrasting theories of coincidences, which I refer to as the ‘normal’ or ‘N Theory’ account and the ‘paranormal’ or ‘P Theory’ account.  N Theory is represented by R.A. Fisher who stated:

. . . for the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us.

Fisher, 1937, p. 16.

The P Theory is represented by Carl Gustav Jung who stated:

Synchronicity – ‘a certain curious principle… [which] takes the coincidence of events in space and time as meaning something more than mere chance…’

Jung, 1951.

These two main lines of thought about coincidences are: 1) the statistical theory that coincidences are inevitable by the normal operation of the laws of chance (N Theory),  and 2) the paranormal theory that coincidences are a special category of experience based on an ‘acausal’ process of synchronicity (P Theory). It can be said from the outset that P Theory and N Theory are mutually exclusive. 

In The Roots of Coincidence, Koestler (1972) discusses the paranormal speculations of scientists, especially those of the physicist Wolfgang Pauli, biologist Paul Kammerer, and psychoanalyst Carl Jung. Koestler developed the suspicion that Darwin’s theory of evolution must be wrong and conjectured, following Hamlet, that there are more things under the stars than are dreamt in our philosophies. The question is, what are those things?

Koestler made no secret of his acceptance of paranormal interpretations coincidences. Koestler’s thinking about the roots of coincidence can be summarised by the assumption that coincidences do not arise by chance from any normal progression of natural events. Koestler liked to collect coincidences that he or his contacts encountered, automatically assuming that something paranormal had occurred. This fallacy is made by almost everybody who attributes a paranormal interpretation to a coincidence.

It is one consequence of probability theory that an event that is very unlikely in the short run becomes inevitable somewhere in the long run. To explore how this happens, imagine that we flip five coins at once. The probability of obtaining five heads is 1/ 32, or .03125.  If we repeatedly flip the five coins ten different times, then the probability of obtaining five heads somewhere in the ten tests is about .27.  In 100 coin flips, the probability of five heads rises to .96, almost a certainty.  However, if we stop flipping the coins anywhere in these 100 tests and ask, what is the  probability of obtaining five heads on the very next trial, we are back at the starting probability of 0.03125 because we have switched from a long-run question to a short-run question.

The illusion of the short run occurs because human memory and attention are limited commodities. In “The magical number seven, plus or minus two”, George A Miller (1956) begins:

My problem is that I have been persecuted by an integer. For seven years this number has followed me around, has intruded in my most private data, and has assaulted me from the pages of our most public journals. This number assumes a variety of disguises, being sometimes a little larger and sometimes a little smaller than usual, but never changing so much as to be unrecognizable. The persistence with which this number plagues me is far more than a random accident. There is, to quote a famous senator, a design behind it, some pattern governing its appearances. Either there really is something unusual about the number or else I am suffering from delusions of persecution.

Miller, 1956, p. 81.

The essential psychological point about Miller’s persistent integer 7 (+/- 2) is that it represents the number of items, thoughts or events one can readily recall in one’s short term memory of the stream of one’s consciousness. It is as if we are living in a resetting ‘bubble’ consisting, at any one moment, of a short run of only 7(+/- 2) things.  When we are bobbing along in our bubbles attending to everyday wants and needs, everything goes swimmingly by, and, at each successive moment, we are content with what we are focusing on.  When, on rare occasion, we come across a peculiar, anomalistic occurrence, and because we still can only perceive the limited contents of the bubble, we conclude that something of true moment has occurred. What we cannot see within our bubble of 7 +/- 2 is the long run of events accumulating over our entire lifetime. If we could see all of the astronomically long sequence of events in one go, we would more easily perceive that anomalistic synchronies or coincidences inevitably must happen.

This statistical perspective leads to the ‘Principle of the Long Run’ (PLR) which maintains that, in a long run of events, coincidences are inevitable.[1]  The PLR holds that coincidences are normal occurrences within any long series of events, hence the name, ‘Normal’ or ‘N Theory’. The principle is  easiest to understand in simple coin, dice or card situation where all the choices are fixed in advance and well defined. It is much less obvious in the more chaotic and complex world of everyday experience where so many different kinds of things are happening all of the time to every single living being. It is possible to show the application of the PLR  to coincidences with a simple thought experiment (Marks and Kammann, 1980; Marks, 2000).  At the end of an ordinary day,  let us assume that it would be possible for a person to recall 100 distinct events.[2] A coincidence requires a match  between one event A and another event B and to calculate the probabilities involved we need to know the total number of PAIRS of events available within a series of 100 (n) single events. This is calculated by the formula:

N x N-1 / 2

From this formula we arrive at a total of 4,950 pairs of events for  a single person in a single day. We tend to remember coincidences for years to come, so that a person can remember all of the important coincidences over the past 10 years, which is approximately 3650 days). [3] Let us assume further that, throughout his/her lifetime, a person has access to 1000 people consisting of family, friends, acquaintances, neighbours, colleagues, etc. This means that there would be a total of 4,950 x 3,650 x 1000 or 18,067,500,000 pairs of events.  That a coincidence collector such as Koestler could find 40 striking coincidences out of 18 billion pairs of events is not so surprising and no reason to invoke P Theory. On the contrary, a good sample of coincidences is inevitable in any population of 18 billion pairs of events.[4]

Of course, when each coincidence individually occurs in each person’s short-run bubble, it seems exceedingly odd. Our memory is not up to the task of paying attention to the entire collection of multiple billions of non-synchronous, non-coincidences. These ‘non-events’ are not perceived as paired in any way compared to the matches that might occur between the names of race horses and Aunt Sally’s pet dog or thoughts about people who then mysteriously telephone a few minutes later  –  these all  make equally striking  anomalous encounters.  We can call this the Principle of Equivalent Coincidences; any coincidence is as good as any other. It is not the probability of any particular coincidence that matters, but the probability of any coincidence over the long run, which approaches 1.0. This, then, is the first root of coincidence, missed by Koestler: the inevitability that, sooner or later over the long run, a thought will match an event purely by chance. Nothing paranormal is happening, just the playing out of chance. There are literally billions of possible combinations of thoughts and events over one’s lifetime and some are bound to produce synchronicities.

Another root of coincidence is the ‘unseen’ or ‘hidden’ cause. A variety of unseen causes can interfere with our interpretations of anomalistic phenomena both as individual perceivers and as scientific investigators. Examples of unseen causes are: sensory cues, non-randomness, deception, tricks and pranks, expectancy, confirmation bias, subjective validation, population stereotypes, hidden knowledge and, finally, feelings (Marks, 2020). This set of factors are always a possibility but, for the present purposes, it will be set to one side. It must be assumed that the possibility of hidden causes is an unwanted nuisance factor that has to be eliminated before the main scientific task of choosing between N Theory and P Theory.

Carl Gustav Jung’s (1969) synchronicity theory focuses on the meaningfulness in the relationship between the two coinciding events, A and B.  It is the striking significance of meaning that promotes the cause of paranormality, especially precognition.  Jung viewed synchronicity as an “Acausal Connecting Principle”, which is “informative, emotionally charged, and transforming the observer’s beliefs or point of view.”  Following a significant anomalous experience, a person tends to ruminate about the event, talk about it, and they cannot let the matter rest until they have worked out a sensible explanation.  Synchronicity between inner thoughts, dreams or images and external events is innately engaging. But here comes the paradox.

The Paradox of Coincidence

According to the PLR, a coincidence is always deemed to be a chance event, no matter how long the odds. Consider another thought experiment: what if paranormal, or P coincidences exist, then they would be mixed in with the normal N coincidences and we would have no way of distinguishing the P type from the N type. By assuming that all coincidences are of type N,  Science is ruling out of existence the very phenomenon that it should be attempting to explain: the possibility that one group of coincidences is caused by an, as yet, undefined non-chance process, process P.

This N Theory assumption that all coincidences are created by chance means that the conventional criterion for evaluating the statistical significance of an observation does not work for coincidences. Null hypothesis significance testing (NHST) could not be applied because we would have no way of knowing the distribution of the population of type P events, e.g. the mean and standard deviation.  The standard criterion for rejecting a null hypothesis H0 is p<.05.  For coincidences, however, this probability can be as low as p<.000000000001 (10-12) or even smaller and the occurrence is accepted as falling within chance levels. 

This double standard in evaluating the statistical significance of coincidences is rather odd and interesting: the more improbable a coincidence, the more it is allowed to be said that it is consistent with the null hypothesis, that the event was caused purely by chance. To avoid miscounting paranormal events as chance events, and thereby defining them out of existence, it is necessary to hypothesise two populations of coincidental events, those due purely to chance, and those that have a putative paranormal source.   

How can we escape this paradox? Imagine a coincidence with an odds of 10 to the minus 12. This is one chance in a million million.  Consider the  possibility that there is an alternative to the normal, null hypothesis, which we can call the ‘paranormal hypothesis’ which assumes coincidences can also have paranormal origin. According to this hypothetical framework, a cross-over point with a Bayes Factor of 1.0 occurs will occur somewhere along the horizontal axis (Figure 1). Let us assume purely for the sake of discussion that the cross-over point occurs with an odds value of 10-12. The hypothesised existence of a separate population of paranormal events places ‘N Theory’ and P Theory on an equal footing and avoids a double standard in hypothesis testing. Of course, it is absolutely impossible to say if this theory correct as, at present, it remains pure speculation.[5]

H0: The null hypothesis – what occurred was purely chance.

H1: The paranormal hypothesis – what occurred was too improbable to be caused by chance and must have a paranormal explanation. 

With this background theoretical analysis in mind, I turn to a consider a specific coincidence experienced by the author. For convenience,  I will refer to as the “Chiswick Coincidence’.  I analyse its structure, meaning, the odds of its occurrence, and possible explanations. A coincidence like the one to be described is both puzzling and remarkable, a contiguity of events that appears to have no causal connecting principle between one another. A coincidence that seems to go way beyond the laws of chance can elicit a strong sense of the paranormal. Although one must never rush to make conclusions, case studies like this provide the opportunity to discover personal meanings and the underlying psychodynamics of the experience. I analyse here the Chiswick Coincidence for the light it may shed on the peculiarities of anomalistic experience.[6] 

The Chiswick Coincidence

23rd August, 2018: At 35,000 feet, on the midday flight from Marseille to Heathrow, I am thinking how to spend the afternoon. Unable to go straight home because an estate agent had arranged to show my flat to a potential tenant, how would I fill this time?  I decide to go for lunch at one of my local haunts on the Thames bank, the City Barge. Set aside for a moment the fact that the estate agent who had arranged the viewing was Chesterton’s, a family firm with connections to the writer GK Chesterton (1874 – 1936),  known as ‘the prince of paradox’.  A few seconds after I made the  decision to go for a pub lunch ‘on the Chiswick bank of the river’, I open my kindle and make a fairly random decision to continue reading ‘The Man Who Was Thursday, a Nightmare by Gilbert K Chesterton (GKC). I flip over the page to see in black and white a description of that very place which, moments previously, I had decided to visit, viz:

“”I think,” said Gregory, with placid irrelevancy, “that we will call a cab.” He gave two long whistles, and a hansom came rattling down the road. The two got into it in silence. Gregory gave through the trap the address of an obscure public-house on the Chiswick bank of the river. The cab whisked itself away again, and in it these two fantastics quitted their fantastic town.”

‘The Man Who Was Thursday, a Nightmare by Gilbert K Chesterton

The correspondence between the free and voluntary thought of going to the pub on the Chiswick side of the river and Gregory’s choice to do the identical thing is particularly striking. This coincidence, like others that I, or close family members, have experienced is multi-layered. I discuss here each of these 7 layers.

Seven Layers of Synchronicity

First layer

The ‘Chiswick Coincidence’ consisted of two elements:

Element 1: Because an agent, Chesterton’s, would be showing my flat to a potential tenant, I decided to go to the City Barge for lunch.

A few seconds later:

Element 2:  Reading the line ‘an obscure public-house on the Chiswick bank of the river’ from a novel by G K Chesterton on my kindle.

Thus, the first layer of coincidence is the fact that the estate agent and the author GKC are members of the same family.

Second layer

The second layer is the fact that the decision to go to the Chiswick riverside pub was followed only seconds later by reading a piece of text referring to a ‘public-house on the Chiswick bank of the river’. My immediate reaction being “Wow!”, “Whoa!” “WTX!” in no particular order. I knew how much text is on my kindle and, seeing these particular lines immediately I began reading, was a complete surprise.

Historical records indicate that The City Barge has existed since 1484 when it was known as ‘The Navigator’s Arms’. Its first appearance in the licensing lists was in 1787 when it was the ‘City Navigation Barge’. As the ‘City Barge’ it was refurbished in 2014.  Historical sources point to at least 5 or 6 pubs on the Chiswick side of the river at the time of GKC’s story. The pub mentioned by GKC could have been any or none of these, perhaps only a figment of GKC’s fluid imagination. Two clues make the City Barge a good candidate however. Photographs of the City Barge from 1910, two years after the publication of TMWWTAN, show Thames barges actually tied up directly outside the City Barge. Also, when the two characters in GKC’s story, Gregory and Syme, leave the pub, they go out by the door and “close to the opening lay a dark dwarfish steam-launch”. This description fits the immediate riverside proximity of The City Barge perfectly.[7]

A kindle is a portable library. Mine is 1.33 GB with multiple books, both fiction and non-fiction, the complete works of Shakespeare, Chaucer, Dickens, Joyce, Austen, Pepys, Swift and Zola.  On the date in question, there were 498 works containing 146,817 pages [8]. With 350 words per page, there were around 50 million words on my kindle.  The odds of seeing the words “public-house on the Chiswick bank” on the first page I opened is around one-in-10 million (10-7).

Third layer

I checked my diary for the days immediately following the date of this event. My diary says that I would be meeting my publisher Robert Patterson to discuss a new book on the paranormal (Marks, 2020).  Was I perhaps on the lookout for anomalistic experience at this time?  If so, I had been presented with a brilliant example.

Fourth layer

The idea of writing the book meant that I would soon be seeking new material.  Although I was at the early preparatory stages when this incident happened, I can imagine no more suitable an illustration for a book on anomalous experience than this very incident. Reflecting back on this period, I see how helpful the coincidence was in resetting my paranormal ‘Belief Barometer’. 

Fifth layer

Enter – or, I should say, re-enter – Martin Gardner.  Martin had contributed Forewords to two editions of my previous book on ‘psi’ (Marks and Kammann, 1980; Marks, 2000).[9]  Martin died in 2010 leaving a huge legacy of 100s of literary and scholarly works with a readership of millions. I have copies of many of Martin’s books including Fads and Fallacies in the Name of Science (Dover, 1957), Mathematics, Magic and Mystery (Dover, 1956), The Annotated Alice. The Definitive Edition. Lewis Carroll (W W Norton, 2000).

In researching TMWWTAN, however,  I made the new (to me) discovery that Martin had written a Special Annotated Edition of TMWWTAN (Gardner and Chesterton, Ignatius Press, 1999). Goose bump territory! How very strange. Discovering this Special Annotated Edition seemed enigmatic and enthralling in equal measure. The three-way connection between Gilbert K Chesterton, Martin Gardner and this author, however, does not end here.

Sixth layer

As Chesterton noted, “hardly anybody who looked at the title ever seems to have looked at the sub-title; which was “A Nightmare,” and the answer to a good many critical questions” (Autobiography, Kindle Locations 1301-1303). Two key themes of TMWWTAN are free will and evil.  The Chiswick Coincidence triggered a change in my stance from disbelieving sceptic to neutral inquirer.  My eyes were opened to the genius of Gilbert K Chesterton, certainly a special writer and TMWWTAN is no ordinary book.It has been rated as one of the greatest works of 20th century literature. To quote from the American Chesterton Society website (

“Gilbert Keith Chesterton (1874-1936) cannot be summed up in one sentence. Nor in one paragraph…But rather than waiting to separate the goats from the sheep, let’s just come right out and say it: G.K. Chesterton was the best writer of the 20th century…The reason he was the greatest writer of the 20th century was because he was also the greatest thinker of the 20th century… What was it he defended? He defended “the common man” and common sense. He defended the poor. He defended the family. He defended beauty. And he defended Christianity and the Catholic Faith.”

Reading that GFK had defended the “common man”, common sense and the poor, was pleasing because these are, what I like to believe, my own values exactly.  Clearly, Gilbert Chesterton made no secret of the fact that he believed in God, prayer and the afterlife, beliefs that I do not share. 

Seventh layer

Like GKC, and he also made no secret of it, Martin Gardner believed in God, prayer and the afterlife. In his autobiography, Martin stated he loved reading “anything by G. K. because of his never-ceasing emotions of wonder and gratitude to God, not only for such complicated things as himself, his wife, and the universe, but for such “tremendous trifles” (as he once called them) as rain, sunlight, flowers, trees, colours, stars, even stones that “shine along the road / That are and cannot be,” (Undiluted Hocus-Pocus.The Autobiography of Martin Gardner, 2013, p. 205).  In digging a little deeper into the two men’s biographies,I discovered thatGKC, together with the Basque philosopher and poet Miguel de Unamuno, were Martin’s two mentors. Martin’s autobiography mentions God no less than 128 times.[10] According to Martin Gardner (2013):

“Just as knowing how a magic trick is done spoils all its wonder, so let us be grateful that wherever science and reason turn they plunge finally into stygian darkness. I am not in the least annoyed because I do not understand time and space, or consciousness, or free will, or evil, or why the universe is made the way it is. I am relieved beyond measure that I do not need to comprehend more than dimly the nature of God or an afterlife. I do not want to be blinded by truths beyond the capacity of my eyes and brain and heart. I am as contented as a Carnap with the absence of rational methods for penetrating ultimate mysteries” (p. 341).

For many reasons, and in completely unexpected ways, the Chiswick Coincidence ‘opened my eyes’. At a seventh layer, I find that the coincidence revealed another synchronicity: the shared values and beliefs of Martin Gardner and a man I could never have met, GKC, author of the metaphysical novel, TMWWTAN.  


I estimate here the probabilities for the synchronicities at each layer, followed by a combined probability estimate. 

Layer 1: The probability that the estate agent and GKC are from a single family is estimated to be 10-3. This estimate takes into account the number of West London estate agencies (500+) and the chance that the agent that I had selected would have a strong familial connection with GKC, the central character in this episode.

Layer 2: The probability that my intention to visit the riverside pub at Chiswick would be followed a few seconds later by seeing the words ‘public-house on the Chiswick bank ’on the first page of my kindleis estimated to be 10-7. This estimate takes into account the huge quantity of kindle text (in excess of 50 million words).

Layer 3:The probability thaton the same visit to London I would meet my publisher to discuss a new book on anomalistic psychology is estimated to be 10-1. This accords with the frequency of such meetings, which is approximately once a year.

Layer 4:Taking into account the fact that no contract for this book existed at this time, the probability thatthe Chiswick Coincidence would be helpful to the book is estimated to be 10-1   

Layer 5:Taking into account of the fact that, before this incident, I knew almost nothing about GKC, the probability thatsomebody I knew and had written forewords to two of my books, Martin Gardner, would also be the very person who had written a Special Annotated Edition of TMWWTAN is estimated to be 10-4.

Layer 6: The probability that lifelong personal values to defend the “common man”, common sense and the poor, I later discovered to be GFK’s values, is estimated to be 10-1 [11].

Layer 7: The synchronicity between the values and beliefs of Martin Gardner and G K Cis estimated to be a certainty. Martin loved GKC’s writing, shared his values and beliefs, and would not have produced an annotated edition of TMWWTAN otherwise.

In addition, it is necessary to consider one of the boundary conditions. Sitting on an aeroplane on a short-haul flight, offers a variety of activities, viz: doing nothing, doing a puzzle, watching a film, listening to music, snoozing,  chatting,  looking out of the window, drinking a tea or coffee, reading a non-kindle item (newspaper, magazine or book), or reading a kindle. I estimate the probability that I would have chosen to read my kindle on this occasion as one-in-ten ( 10-1 ).

The combined probability P of the seven synchronicities and the boundary condition is:

 P  = 10-3 X  10-7 X  10-1 X  10-1   X  10-4   X 10-1  X 1 X  10-1   = 10-18

= one in 1,000,000,000,000,000,000 

i.e. one in one quintillion (a million, million, million) [12] 


One quintillion is so huge, it is difficult to comprehend without a concrete illustration. As another thought experiment, imagine that we convert this abstract number into rice grains, one quintillion rice grains. The rice grains are all husked and light brown in colour. Completely at random, imagine that I replace one of the rice grains with a grain-sized gold nugget.  There are 15,432 grains in each kilogram. The total weight of the one quintillion rice grains is 64,798,909,802,406 kilograms. With a density of 753 kg/m3, the rice occupies a volume of 86,054,329,087 cubic metres. Comparing this to the world’s largest mountain, this amount is one-and-a-half Mount Everests. For the last 20 years, global rice consumption has averaged close to 54 kilograms per person (Statista, 2019). The enormous rice mountain created by one quintillian grains would feed the entire world population of 7.7 billion for 156 years. The odds of finding a single grain of gold inside this enormous mountain would be identical to the odds of my Chiswick Coincidence. These odds are so astronomical, one must consider the possibility of a paranormal explanation. Not to do so would be irrational and contrary to science.

Explaining the Chiswick Coincidence

How might this remarkable 7-layered coincidence, together with its impact and meaning, all be explained?  Let’s consider the explanations that are available from each side of the theoretical divide.

Hypothesis 1 – N Theory Explanation: Coincidences are bound to occur every once in a while, purely by chance.

From the perspective of N Theory, I give the first type of explanation. The nugget of the Chiswick Coincidence lies within Layer 2:

Event A: choosing by free will to go to the City Barge for lunch.

Event B: choosing by free will to read, only moments later, a story, I would soon discover, that contains an incident about a  ‘public-house on the Chiswick bank of the river’.

When considered independently, neither event is in any way extraordinary. Only their near simultaneity appears extraordinary. If I had read the passage a few months, weeks or even days previously or sometime later, I would have noted that I knew just such a place but would not have blinked an eyelid.  Any Londoner is familiar with the experience of coming across familiar places in novels or movies.

It is necessary to consider the possibility of a hidden cause, something that might create the illusion of synchronicity when it isn’t really there. One possibility is that GKC may have been frequently mentioning things in and around Chiswick. In this case the coincidence might not be so odd after all.  It is possible to test this hypothesis relatively easily. It is said that Chesterton was one of the most prolific writers of all time. He wrote around 80 books, several hundred poems, 200 short stories, 4000 essays, and several plays. I downloaded the Delphi Collected Works of GK Chesterton onto my kindle. Using the kindle search function, I found that there only 7 occurrences of the word “Chiswick” in GKC’s Collected Works. This fact makes the Chiswick Coincidence seem even odder than before.

Another possibility that must be considered is that I had already seen the crucial passage on a previous occasion. This possibility can be safely eliminated for two different reasons. Firstly, if I had already seen this passage, I would already have noticed the connection between one of my favourite riverside haunts and GFC’s mention of it. In this case, seeing it for a second time would not have seemed terribly remarkable. Secondly, a kindle automatically remembers the point reached at a previous reading and obligingly opens the selected book at that page.

The ultimate sceptical explanation is possibly the most accurate. It says that coincidences are just – coincidences!  A coincidence is a coincidence is a coincidence; a random, chance kind of thing. Something similar to the Chiswick Coincidence is occurring with someone somewhere almost every second of the day. When this extremely striking kind of coincidence occurs, it is bound to attract the experiencer’s attention. It is purely the wheels of chance turning and nothing else.

Hypothesis 2 – P Theory Explanation: Reverse causality by unconscious reading of the text triggers the decision to visit the pub on the Chiswick side of the river.

What of a paranormal interpretation? It is essential to air all possible explanations and the P Theory warrants a fair hearing.  The two key elements of the Chiswick Coincidence remain :

Event A: deciding by free will to go to the City Barge for lunch.

Event B: deciding by free will to read, only moments later, a story, which contains an incident about a ‘public-house on the Chiswick bank of the river’.

What about the possibility of retro-causality such that Event B occurs immediately before Event A?  However fantastic it may seem, it is only fair that it be stated. The P Theory explanation goes like this: I read the part of the story about the Chiswick pub by an unconscious process of clairvoyance, clairvoyantly seeing the text about a ‘public-house on the Chiswick bank of the river’ inside my kindle.  Reading this text at an unconscious level triggers my decision to go to the City Barge for lunch. Afterwards, at a conscious level, when I switch on the kindle and actually read the text, I feel a sense of wonderment and surprise. This is no coincidence at all – reading about the Chiswick pub naturally and logically led to my plan to visit it.

If one is open to psi as a scientific possibility, there should be no problem in accepting the P Theory explanation.[13]  The P Theory nails it. If the sceptic demurs that there is no evidence for clairvoyance, unconscious perception or reverse causality and it just cannot be so, the P Theorist might retort: “Normally, yes, but on this occasion all three happened.” There is no rational way of resolving the matter; which interpretation one accepts rests entirely upon subjective judgement.


On a homeward journey, involving multiple free choices, a striking coincidence happened (Figure 2). The laws of chance suggest that the odds against the Chiswick Coincidence are around one-quintillion-to-one. Both ‘N Theory’ and ‘P Theory’ interpretations remain logical possibilities. There can be no definitive method of proving which explanation is the correct one. This incertitude requires a neutral stance and a degree of humility about one’s reaction to striking anomalous experience.[14]

My search for a scientific explanation was matched by an equally compelling realisation that there might not be one. Which interpretation is true cannot be decided by reason. Only personal preference – based on one’s a pre-existing bias – allows one to reach a definite conclusion. This insight led me to question my lifelong mental habit of scepticism.  In one ultimate act of scepticism – I became sceptical of scepticism.



In addition to the Chiswick Coincidence, I describe elsewhere four other striking coincidences over my lifetime.[15] It is enlightening to compute the odds that all 5 of these events could happen to the same individual. To determine the probability of five independent events, A – E,  we multiply the individual probabilities together: 

P(A and B and C and D and E) = P(A)×P(B) x P(C)×P(D) x P(E)  

The five coincidences are as follows:

A) An obscure public house on the Chiswick bank of the river: P=10-18 = one chance in 1,000,000,000,000,000,000  = one in one quintillion (a million, million, million)

B) Coincidence or Luck?: P = 10-10  = one chance in ten billion.

C) Citizen 63 – Marion Knight: P= 4.5 x 10-10  = 4.5 chances in ten billion

D) The Case of the Flying Horseshoe:  P = 1.3 x 10-12 = 1.3 chances in a million, million

E) Under the Wallpaper:   P = 5.08 x 10-9  = 5 chances in a billion

By simple multiplication, the combined probability of these five independent events is :

P =   10-18   x   10-10   x   4.5 x 10-10  x 1.3 x 10-12    x 5.08 x 10-9

P = 3 x 10-58 

A probability of 3 x 10-58 is one of the smallest probabilities imaginable.  Yet according to the widely accepted Principle of the Long Run, of N Theory,  there is nothing extraordinary here. To try to make sense of such astronomical odds, we may speculate about the possibility of a second source of coincidence anomalies with a paranormal origin.

Synchronicity and meaning are existential processes with an ancient biological role. Synchronicity binds thoughts, feelings and behaviour with meaning. The standard scientific explanation based on normal probability theory adduces the statistical Principle of the Long Run which dictates that improbable coincidences are inevitable. However, an alternative hypothesis that there is the possibility of psi cannot be ruled out. There is no rational method for determining which interpretation is more likely to be correct. We each must draw our own conclusions.


Browne, L. (2017). The many faces of coincidence. Imprint Academic. Kindle Edition.

Carroll, L. (2000). The annotated Alice. The definitive edition. Introduction and notes by Martin Gardner. New York and London: WW Norton.

Chesterton, G. K. and Gardner, M. (1999). The man who was Thursday. A nightmare. With annotations by Martin Gardner. San Francisco: Ignatius Press.

Chesterton, G. K. (2006). The autobiography of GK Chesterton. San Francisco: Ignatius Press.

Diaconis, P. and Mosteller, F. (1989). Methods for studying coincidences. Journal of the American Statistical Association84(408), 853-861.

Feinberg, G. (1975). Precognition—a memory of things future. In L. Oteri (ed.), Quantum physics and parapsychology. Parapsychology Foundation. Pp. 54—64.

Fisher, R.A. (1937). The design of experiments (2nd ed.). London: Oliver & Boyd.

Gardner, M. (1956). Math, magic and mystery. New York: Dover.

Gardner, M. (1957). Fads and fallacies in the name of science. New York: Dover.

Gardner, M.(2013). Undiluted hocus-pocus. Princeton University Press. Kindle Edition.

Martin, B. (1998). Coincidences: Remarkable or random? Skeptical Inquirer, 22(5).

Hand, D. J. (2014). The improbability principle: Why coincidences, miracles, and rare events happen every day. Scientific American/Farrar, Straus and Giroux.

Jung, C. G. (1951). Foreword to Richard Wilhelm’s translation of the I Ching. UK: Routledge & Kegan Paul.

Koestler, A. (1972). The roots of coincidence. New York: Vintage Books.

Marks, D.F. and Kammann, R. (1980). The psychology of the psychic. Foreword by Martin Gardner. New York: Prometheus Books.

Marks, D.F. (2000). The psychology of the psychic (2nd ed.). Foreword by Martin Gardner New York: Prometheus Books.

Marks, D.F. (2020). Psychology and the paranormal. Exploring anomalistic experience. London: SAGE Publications.

Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological review63(2), 81.

Statista (2019). Global use of rice per capita from 2000/2001 to 2018/2019 (in kilograms per year). Available at:

Thalbourne, M. A. (2006). A brief treatise on coincidence. Unpublished manuscript. Online at http://parrochia. wifeo. com/documents/coincidence. pdf. Accessed 06 December 2010.

[1] The Principle of the Long Run reappeared in a new guise as the ‘Law of Truly Large Numbers’ in a Science paper by Diaconis and Mosteller (1989)  See also: ‘The Improbability Principle’ (Hand, 2014).

[2] The estimate of 100 is not empirically based, but simply a convenient, round number. The true figure could be lower or higher.

[3] We can compute the number of pairs over a single person’s adult lifetime of 60 years 4950 x 60 x 365 = 1,084,050,000 pairs + 15 x 4950 = 74,250 extra pairs for leap days giving a total of  1,084,124,250, i.e. slightly more than one billion pairs. If a coincidence is a one-in-ten-million event, then we can expect around 100 striking coincidences in one lifetime. At one-in-100-million, however, there would be only 10 per lifetime, and at one-per-billion, only one. These figures are based on the assumption that there are 100 discrete events per day. If this estimate is changed, then the above figures would need to be revised proportionately.

[4] In Irrationality: the enemy within (p. 227) Stuart Sutherland misapplied the 18 billion figure as follows: “Arthur Koestler sought to establish the truth of the paranormal by pointing to fifty coincidences that had occurred in his life, which he claimed could not be given any normal explanation, but Marks and Kammann point out that in a lifetime he would have been exposed to over 18 billion pairs of events: it would be most unlikely if some of the members of a pair did not match.” As computed in footnote 17 above, there should be only about one billion event pairs in a single lifetime.

[5] Laurence Browne (2017) proposes four categories —a) random chance; b) natural causality; c) supernatural causality; d) synchronicity—to represent the four main ways in which coincidences are customarily explained. In the model of Figure 1, we include only types a and c. There is no evidence that type d is necessary as a separate category from type c.

[6] I freely acknowledge some readers may well view my ‘Chiswick Coincidence’ with scepticism. See: ‘One person’s coincidence can be another person’s yawn’;

[7] The City Barge is only a 10-minute drive from Bedford Park, the “queer artificial village” of  ‘Saffron Park’ in GKC’s novel.

[8] With the settings on the kindle as they were at that time, there are 4-5 kindle pages to every printed page.

[9] Martin Gardner (Foreword to the Second Edition, Marks, 2000) wrote: “Twenty years have passed since The Psychology of the Psychic, by David Marks and Richard Kammann, was published…Much has happened on the psi frontier since then. Kammann died in 1984…Marks has so thoroughly revised and expanded the text that it is almost as if he has written an entirely new volume. It will rank as one of the strongest and best exposés ever directed at the more outlandish claims of parapsychology”(p. 13).

[10] By comparison, Chesterton’s autobiography mentions ‘God’ 62 times.

[11] I share GKC’s values as listed but not his Roman Catholic religious beliefs.

[12] A quintillion is cardinal number represented by 1 followed by 18 zeros (US) and by 1 followed by 30 zeros (UK). Here I use the US definition.

[13] Physicist Gerald Feinberg (2005) sees no incompatibility between modern physics and precognition:  “Instead of forbidding precognition from happening, [accepted physical] theories typically have sufficient symmetry (between past and future) to suggest that phenomena akin to precognition should occur. . . . Indeed, phenomena involving a reversed time order of cause and effect are generally excluded from consideration on the ground that they have not been observed, rather than because the theory forbids them. This exclusion itself introduces an element of asymmetry into the physical theories, which some physicists have felt was improper or required further explanation. . . . Thus, if such phenomena indeed occur, no change in the fundamental equations of physics would be needed to describe them.

[14] Michael Thalbourne (2006) dismisses sceptical explanations based on chance “as a bottomless pit, able to swallow up each and every coincidence that does not already have anormal explanation.”The fact is, in regard to this coincidence, there is no fool-proof method to say whether the P Theory of the N Theory interpretation is correct. It comes down to making one’s own subjective evaluation.

[15] See:

Published by David F Marks

Author, editor, psychologist.

2 thoughts on “The Paradoxical Nature of Coincidence


    David, synchronicity and following my intuition have had such a life-changing effect on my life. Thought you might be interested in the above post and links within it that detail some of the incredible synchronicities in my life. If it doesn’t mention the one in India and you are curious, I’ll relate it to you at another time, but I think it is mentioned either in the talk or one of the posts. Thanks for your link.

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