We are deceived by our limited impressions of the world’s size. It is unimaginably and frighteningly vast. Yet its people seem to come together in time and space with a baffling frequency that shrinks the world to a more comforting size. If we ignore all the sensible reasons that make the frequency of coincidences mathematically predictable, their stories transmit a strong sense of inclusive human connectivity, justify our existential significance, and validate our longing for individuality.

In March, I published a book about the frequency of coincidence stories, called Fluke. In the two months since, I’ve become the beneficiary of a great many stories from readers detailing the flukes in their own lives. A vast majority of these stories are about chance meetings of neighbors or acquaintances while traveling in foreign cities. Some are meaningful flukes that test our belief in chance. There are dreams that come true, coincidences involving familiar objects that appear in unexpected places, and extreme stories of good and bad gambling luck. Most of these seem improbable, but when considered mathematically, they are not really so unlikely.

And then there are the one-of-a-kind flukes that happen against staggering odds. Some are funny, some sad, some disturbing, and some suspiciously exaggerated secondhand yarns. All provoke clashing thoughts of chance and destiny.

Here are four wondrous stories, chosen from the many and ranked by increasing odds against, that shrink the size of the world.

* * *

At a bookstore reading one gentleman in the audience by the name of Ted told me this coincidence story: In 1989 Ted was traveling south from Philadelphia with a change of planes at Reagan National Airport near Washington. Boarding time for his second flight was already delayed 30 minutes. Then the airline announced that it would be delayed another hour because maintenance was waiting for a new flap motor to be delivered and installed at the gate. He turned to a man standing next to him, someone he had never met, and expressed his concern that the plane would take off without prior functional testing. Ted told this stranger that at his company, DuPont, functionally testing a repair with passengers on board would never happen.

“Oh, you work at DuPont?” said the stranger pointing to the nearby payphones. “I was just trying to reach a guy at DuPont who left me a voice message. I’m trying to return his call. I wonder if you know him?”

At that time DuPont had 140,000 employees, so Ted was quite doubtful, but he politely asked for the name. When the stranger answered with Ted’s own name, he was dazed and animatedly called out, “You are speaking to him!”

* * *

Soon after the launch of Fluke, I received a handwritten letter from the Petersburg Federal Correctional Institution from an inmate, along with other documents, including a long letter to the deputy attorney general of the U.S. Department of Justice. This fan was the owner and founder of a social media mobile content and technology company based in Washington. He had been sentenced to 30 months in prison for failing to pay the Internal Revenue Service a large amount in owed payroll taxes. He pleaded guilty, so we might think that that closes the case. Maybe. He asserted in his letter that during a two-year period, out of 37,400 businesses in the Washington metro area, the only two that were prosecuted were both top Republican Party social media firms; he asked me to answer to the question of the chances of that happening. “I was wondering if you would take a look at the math enclosed,” he wrote, expecting that I would corroborate his belief that he was on “a hit list” as a result of having “embarrassed the President.”

* * *

During one of my radio book tour interviews, a caller told a story involving a fluke of precise timing. Near his childhood home in Omaha, Nebraska, is a great oak tree with one fat limb protruding almost perpendicularly from its trunk. For many years a rope swing hung from that limb. One day when he and his girlfriend were swinging together, he felt a short tremor and heard a sharp cracking sound that made him worry that the heavy branch would soon fall. It didn’t, but he never swung from that swing again. Once a year he would return to visit his parents in Omaha and pass that tree. On an unusually windy afternoon 40 years after that last swinging moment, he once again returned home. The swing was long gone. Only the remnants of rotten rope tatters remained. He passed that oak, looked up, and saw a crack widen, just before the limb came crashing down. “I just looked at the tree. Didn’t touch it. Bet you can’t tell me the odds of that,” he declared.

* * *

One that happened to me was recalled too late to make it into Fluke. Thirty years ago I invited to dinner at my Vermont house two people I had never met, a graduate student from Italy and a professor of literature from India. A college friend, whom I had neither seen nor heard from in many years, dropped by impromptu. He too stayed for dinner. Well into the conversation, when my friend felt it was time to reminisce about our mutual old friends, a street name in Brooklyn came up. “I’ve been there,” said the student. When the address and apartment number of our old friend surfaced, the student, without any indication of surprise on his part, blurted, “Oh, that’s the address of a friend of mine too. I was there yesterday.” And he was!

* * *

Is it possible to know the odds of these coincidences?

On the surface, the first story seems to hinge on Ted being 1 out of a pool of 140,000. No doubt extraordinary. However, the extraneously large number of DuPont employees unintentionally induces an inflated impression of the odds. As with other coincidences, the chances improve with the details. DuPont did business with ChemDesign, the company for which the stranger worked. Ted later told me that he traveled by air “on average a day or two, three out of four weeks a month.” That translates to spending roughly one whole day a month in an airport lounge. Yes, the chances improve with this information, but without speculatively guessing some numbers we cannot assign a realistic chance of Ted standing next to the stranger at the airport.

The second seems to have relatively impressive odds of 18,659 to 1 against happening. But it only seems that way because we are discounting the possibility that the prisoner’s conspiracy contention is correct. If it is, the prisoner’s story is not a coincidence at all. It would have an apparent cause, not dictated by chance alone. It could have been rigged to definitely happen. That does not, however, counteract the fact that tax evasion is illegal.

One approach to the third story is to consider all the short time intervals in all those 40 years when the storyteller did not visit the tree, those times when the branch did not fall. Suppose that the branch came down in a 10-minute interval surrounding his visit. There are 2,102,400 10-minute intervals in 40 years. So the odds are 2,102,399 to 1 against, better than a royal flush of, say, spades. Like royal flushes of spades, very unlikely events do happen.

The last story is impossible to deconstruct without knowing anything about the hidden variables that connect the student with the friend he visited in Brooklyn. This one has extraordinarily high odds against, a veritable head-scratcher. Not every story can be reduced to probability theory. Many of the best cannot, thanks to the many splendid wonders of the world’s hidden variables.

These seemingly unlikely events happen because of the vast number of experientially available possibilities. Did they happen by chance? Or do we use chance as the excuse for what we do not know? The world is mind-bogglingly vast yet made small by our coincidence stories that expand our neighborhoods and make us question whether our collisions are chances or destinies.