It’s No Coincidence: Rarity Is Subjective

I have in my hand—you’ll have to trust me—a certain blade of grass. He is my favorite. The titling curvature of his blade can only be described as rakish, although other tasteful persons have used the word jaunty. I call him “Bob.” The way I found him is a strange story.

A baseball was thrown by some guy, unknown to me, on the Sheep Meadow. The ball was meant to be caught by a second guy, but he missed. The ball rolled near to where I was standing, ultimately stopping smack on top of Bob.

What a remarkable coincidence!

Of all the parks and all the fields and all the blades of grass in the world, that baseball stopped right on top of Bob. My Bob!

What are the chances?

They can be calculated, but only if we keep in mind that all probabilities are conditional on certain premises, or assumptions. This means I cannot ask, “What are the chances the baseball would land on Bob?” without also adding some form of background knowledge.

For instance: suppose I said, “What are the chances given all the parks and all the fields, etc.?” This uniquely specifies a set of conditioning information or background knowledge. Well, perhaps uniquely. As long as I can unambiguously define what I mean by “all parks, all fields, all blades of grass” then I am in business.

To calculate the chance that a thrown baseball lands on Bob is then an exercise in counting. Must be at least as small as one in hundreds of trillions, probably a lot smaller. There is, after all, a lot of grass in all those parks. Let’s call it a conservative 1 in 1015, which is a 1 with fifteen zeros after it.

My finding Bob was thus miraculous, or nearly so. Given I assumed that all those parks, etc., I can conclude, since the chance was so minuscule, that Bob and I were predestined to meet. There must be something magical about him.

On the other hand, maybe it isn’t so miraculous. If I substituted, “given just the blades of grass on the Sheep Meadow,” then the chance I would meet Bob is still small—must be millions of blades of grass on the field—but substantially larger than before.

It’s even larger if I change the question entirely and ask, “What are the chances that a thrown baseball lands on a blade of grass given the field, etc.” The probability is now nearly certain, almost 1. It isn’t exactly 1, because the ball could have rolled onto a blanket, say, or onto a rock (there are big ones on the Sheep Meadow).

The probability of that ball finding Bob is then anywhere from near 0 to near 1, from surprise to banality, depending on what information I supply, on what premises that I feel are important. If I’m determined to find cosmic significance in meeting Bob, then I’ll opt for “all parks, all fields” information. But if I care not about leafy green matter, then I’ll opt for the “any grass” interpretation.

Surprise, or coincidence, is thus in the eye of the beholder.

Persi Diaconis (an old advisor of mine) and Frederick Mosteller struggled to find a definition of coincidence: “a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.”

We now know that “surprising” is a matter of interpretation: a given event can be surprising or not by fiddling with the conditioning information. But what about the emphasis on a lack of causality?

If we knew of some mechanism that directed baseballs to land on blades of grass—perhaps there is a form of subatomic force that causes the threads on balls and the chemicals in the grass to mutually attract—then we wouldn’t be that surprised when a ball hit a blade of grass.

That information about causality naturally forms part of the background information. I can include the physical thread-grass bonding theory into the conditioning premises. Or I can choose to leave it out. As long as I made no mistakes in calculation, the probabilities resulting from both assumptions are correct. That is, I can hold to my low probability, knowing it is true, and I can then revel in the probability’s smallness and say it is meaningful.

But if another person questions why I left a known physical theory out, then in order to convince him that finding Bob was a surprising coincidence, then I have to persuade him the physical theory is irrelevant in this particular case.

Knowledge of causality, then, is just one more piece of background information. Diaconis and Mosteller were right to emphasize it, though, because it reminds the discoverer of a “coincidence” that the burden is on him to show that possible causal mechanisms are irrelevant or nonexistent for this event.


  1. Eric Anderson

    Well this is very surprising . . . no wait, perhaps it isn’t . . . 🙂

    I think one of the keys in determining whether something was in fact improbable (i.e., properly defining the conditions, as you say), is understanding whether there is some independent significance to the event — independent of the event itself. For example, if you had scoured the field and located Bob before the game, and then had watched the game closely to see whether a ball would land exactly on Bob, then it becomes an improbable event (though perhaps not hugely so, depending on the the size of the field, Bob’s location on the field, the number of balls hit/thrown, etc.). It is the finding and identifying of Bob beforehand that is adds independent significance apart from the subsequent event.

    I once heard a talk show on NPR where the guests were debating, among other things, improbability. One of the guests was arguing, in essence, that we should never be surprised about rare events because improbable things happen all the time, and therefore probability calculations are largely meaningless. For example, said he, thinking himself very clever: “What are the odds that of all the people in the world you and I would be ones on this show at this very day and hour?” I don’t recall whether the other guest gave a good response, but I thought to myself: well, given that you are both in the relevant field, given that you both received an invitation from NPR to appear on the particular show at the particular time, given that you both accepted the invitation, it is in fact extremely likely that you would both show up at that very day and hour, even approaching 1 (there could have been an accident on the way or some other unforeseen circumstance, but most guests do appear and most shows do go off as planned).

    A very different situation, and very different calculation, would be if we identified beforehand two random people in the world and asked, for example, what the odds would be of those two people showing up on a particular talk show a year from today. Yes, that would be unlikely. But even in that case the odds would be dramatically reduced by subsequent intervening acts as soon as, for example, NPR placed those two individuals on the list of potential guests and started the ball rolling on producing the talk show . . .

  2. Eric Anderson

    Apologies for the second comment in quick succession. What are the odds I would do that . . .

    I appreciate your post because, coincidentally, this past week I’ve been working on a little project with my middle-school-aged son to get him thinking about probabilities.

    We took Scrabble letters (one each of the 26 letters in the alphabet) and put them in a jar. I have him shake them up and then blindly draw one letter. We write it down and then he puts the letter back and shakes up the jar and the process repeats. We did a series of two-letter, three-letter, and four-letter draws (we’ll eventually go up to ten and then probably quit there). The odds of any particular combination are 1 in 676, 17576, and 456976, respectively. However, as I explained to my son, there are lots of words in the English language with the right number of letters, so we have to take that into account in determining the real probability of drawing a valid word. For example, there are numerous four-letter words in English (no pun intended), so this fact dramatically increases the odds of drawing a valid word in any given try. And sure enough, when we got to the four-letter draw, the first four letters he drew were R-O-D-E, a valid word. As expected, though, it quickly deteriorated from there, and in several subsequent draws nothing but nonsense resulted.

    With three-letter draws, we got one valid word “CAB” out of 20 draws, roughly the ratio we would expect, given 17576 potential combinations and over 1000 three-letter words in English.

    Anyway good stuff. I sometimes wish the teachers at school would give hands on lessons like this, but hopefully this lesson will stick and I won’t lead him too far astray . . .

  3. Briggs

    Eric Anderson,

    My heart soars like a hawk to read your responses. Love the NPR story.

  4. TMI

    Another fan of Douglas Adams.

  5. @ Eric Anderson: So your mission in life, should you choose to accept it, is to tuck the details of the scrabble letter experiment away in your mind until someday far in the future whilst riding in a taxi with your son, and you turn to him and say, “Remember those first three word and four word responses from our letter game so long ago? What would you say are the chances . . . . . ?”

    Great examples. Thanks. Made my day.

  6. gcb

    A related phenomenon (I think) is when your personal reality avoidance device (you know, the one with the white ear buds and the logo that looks like a piece of fruit with a bite taken out of it?) suddenly goes on a “kick” for a particular artist – say it plays three Willie Nelson songs in a row. (Now, on my PRAD, that would be a neat trick, since there are no such songs to be found, but I digress.) Anyway, the first thing to consider would be the number of Willie Nelson songs relative to the total songs on the device as a whole – if you only have the collected works of Willie Nelson loaded, then playing three such songs in a row is not an unlikely event.

    But even then we are considering the specific instance of Willie Nelson songs – perhaps you have a hundred artists represented on your PRAD, with ten songs from each. In that case, the likelihood of running into a three-song stretch by any particular artist still seems to be rather rare at first, until you consider that it didn’t have to be Willie Nelson whose three songs popped up in a row and made you wonder if your device was celebrating a new-found love for all things country. It could just as easily have been three Lady Gaga songs in a row, assuming your playlist was “born that way”. We only noticed that it was three Willie Nelson songs in a row because that’s what this instance was.

    Additionally, due to the fact that most people don’t just listen to three songs in a sitting (but let the infernal device blast their eardrums until they’re deaf or the batteries die, whichever comes first), then if you make your string of events long enough, you’re bound to get three in a row eventually. And, of course, that’s what stands out, not the many many times where it didn’t play three songs in a row by the same artist.

    Just my $0.02…

  7. @ Eric Anderson:

    It was most rude of you to say that any two people in the world, picked at random, could’ve done just as well as NPR’s carefully selected guests. Never mind that you’re entirely correct (messin’ with ya).


    Thank you for not mentioning one of the oldest jokes around, which features a tattoo artist, a natural redhead, and a drunk off the street who loudly declares, “And the one in the middle is Willie Nelson!”

    @ the world:

    Serious question — would you call the presentation of the background information, particularly when irrelevancies are included, confirmation bias? “Don’ know much about biology…”

  8. Nomen Nescio

    Dear Mr. Briggs,
    Your description of Bob leads me to the almost certain conclusion that you found my favorite blade of grass. It sounds exactly like him, though he was going by the name Rob when I last saw him. Please tell me he is okay and wasn’t hurt in the collision with a baseball. I haven’t seen him in days and was worried he was lying in a hospital somewhere.
    So, what are the chances you and I have the same favorite piece of grass? Gotta be astronomical, right?
    Best Regards,

  9. Rich

    Would this be related to the “Prosecutor’s fallacy” then?

  10. Smoking Frog

    I’ve long wondered whether synchronicity could be due to causal relations which are beyond our ability to notice. For example, you’re in a distant city and you run into someone you know from either your own city or some other city. Naturally, there could be an easily understood explanation, e.g., trivially, that the two of you work in the same field, and are attending a conference in that field. But if there is no explanation like this, there might be an explanation so complex that you’d never think of it, or perhaps that you could not even grasp. In other words, there may be causal processes and patterns in society of which we are unaware.

  11. @ Smoking Frog:

    “In other words, there may be causal processes and patterns in society of which we are unaware.”

    How totally, freaking unscientific is that? I can’t hear you, I can’t hear you, I can’t hear you, I can’t hear . . . . . . . . .

  12. Gary

    My father, born and raised in Rhode Island, once attended a golf tournament in Florida. He pointed out to my mother a man standing a few feet ahead of them in the crowd, saying, “I went to high school with that fellow forty years ago and haven’t seen him since we graduated.” Talking to the man confirmed the truth of my fathers statement. Which was more improbable: that my father should meet up with a long lost classmate 1500 miles from home or that he recognize the guy just from the back of his now balding head?

    The family jokingly refers to these occasional coincidences as “Rhode Island moments.” I cite this story because there are multiple pieces of background information one must contemplate to answer the question. Is there a systematic way of conceiving of and judging the importance of relevant pieces of background information, since one cannot know them all?

  13. JH

    The ball rolled near to where I was standing, ultimately stopping smack on top of Bob.
    What a remarkable coincidence!

    It no coincident! It’s a sign. The universe is trying to tell you something. ^_^

  14. txslr

    Back in graduate school I took a class in Philosophy of Science in which we learned about the Paradox of the Ravens. I recall (always a dangerous thing) that the Bayesian approach to the Paradox led to a definition of “surprisingness” derived from Bayes Theorem.

    Now I will have to dig around in my attic to find my old class notes.

  15. max

    I heard it as the lottery problem. With big lotteries they have with a 1-in-300 million chance to win, you can figure the odds of picking the numbers on a winning ticket to be so small as to be effectively zero. Probability that low is accounted to be impossible (especially by scientists who wish to make political points). Thus we know that no one ever wins the lottery. So all the people who claim to have won the lottery are liars because it is impossible to pick a winning lottery ticket.

    Or to put it another way, there are over 6 billion people on the planet earth. If there is a one-in-a-million chance that something will happen to one of them, that means there are over 6,000 people it it will happen to.

  16. ed

    As an engineer,

    Baseball is played in a set of two fields, a field of grass inside a field of gravity.

    Why can’t statisticians grasp the basic mathematical concept of sets?


    ……of course, none of this applies at the sandlot!

  17. My favourite coincidence, and I have quite a number:

    My family migrated from UKLand to Australia in 1965 when I was 14. Within 2 weeks of our arrival, we moved out to Sunbury, a very small town 23 miles from Melbourne. On our second night there, two young blokes knocked on the door and asked: “Are you the people from Warwickshire?”

    One, David Kemp, lived two doors down the road from us in Nuneaton when I was 3 years old. I used to watch Popeye on his parents’ TV. We left that address before I turned 4.

  18. Smoking Frog

    49erDweet said: How totally, freaking unscientific is that? I can’t hear you, I can’t hear you, I can’t hear you, I can’t hear .

    I can’t tell if you’re kidding, serious, or what.

  19. Smoking Frog

    Pompous Git:

    In the 1960s, the Australian government was paying people to immigrate, and if I recall correctly, this included help with buying a home, so I wonder if Sunbury was a place where they just happened to be putting immigrants. This – both the paying and the possible “site selection” – would increase the probability of what happened to you.

    I’m in the US, and I never even remotely heard of anyone who took the offer. Perhaps things were different in the UK, which also would increase the probability.

    BTW, I recall seeing a newspaper ad making the offer and saying that you had to be white to qualify.

  20. Smoking Frog

    Yes, we were ten pound poms; or at least my parents were — the children travelled free. The place the immigrants were being put was the Broadmeadows Migrant Hostel, a most unsalubrious place. My parents found a house to rent at Sunbury that we could afford; it had been condemned as unfit for human habitation and therefore was very cheap. That enabled us to save and purchase a house a year later.

    David Kemp had arrived via Sydney on an aeroplane expecting to settle in Newcastle, in New South Wales, while we had arrived via Melbourne, Victoria by ship. Dave had then moved to Melbourne and looked for somewhere to board. His friend Terry’s parents living in Sunbury were advertising for a boarder; that was how he met Terry. It was a circuitous path.

    Sunbury still only had a population of 3,100 when I departed in 1969. That’s people; there were many more sheep.

    Yes, the UK economy was in disaster mode. We were rats leaving a sinking ship.

    You recall the ads correctly; it was called the White Australia Policy.

  21. Matt

    Serious question — would you call the presentation of the background information, particularly when irrelevancies are included, confirmation bias? “Don’ know much about biology…”

    –Human Person Junior, Jr.

    Are you really truly absolutely certain that they’re irrelevancies?

  22. Smoking Frog

    Pompous Git: Thanks for the reply! I didn’t know the word pom, so I looked it up. Got more than I expected – there are competing theories.

    Yankee is an interesting case in that regard; there are something like 15 theories. The one that’s considered the most likely is that it’s an Iroquois corruption of anglais, but the one I like best is this:

    In the 1600s there was an eccentric (or possibly mentally ill?) farmer in Massachusetts who had a habit of inventing words. He invented yankee as an intensifier(?), e.g., “That’s yankee good soup!”, much as one might say, “That’s damn good soup!” – and it caught on.

  23. World’s Best Pickup Line

    Guy: Do you believe in coincidence?

    Gal: Yes, I guess so.

    Guy (snapping fingers and looking amazed): So do I!!!!!!

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