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.