Impossible Things before Breakfast

Why Rarities are a Dime a Dozen

A few days back, BASTIAT’S WINDOW explored the dark origins of mathematical statistics. To a large extent, statistics (applied probability theory) was created by three famous eugenicists, hoping to give a scientific aura to their racism and snobbery. Today, we’ll focus on the more joyful side of statistics.

In the 1983 film, Local Hero, MacIntyre, a young American oil executive, asks Ben Knox, an elderly eccentric living on a Scottish beach, to name the most amazing thing he ever found washed up on the shore. “Impossible to say,” Ben responds. “See, there’s something amazing every two or three weeks. I’ll let you know the next time.”

Last week, a friend posted a pair of photos, (shown above) from a married couple in China. One had the wife, Xue, posing in front of a monument in Qingdao, with her husband, Ye seeming to photo-bomb her, posed in a distinctive stance. The other picture showed Ye at the same moment in a closer-up shot. In both photos, three other tourists were seated in the distance, conversing. The side-by-side photos went viral on social media because, you see, they were taken over ten years before Xue and Ye met. Ye happened to thumb through Xue’s old photo album and was stunned to see himself in the background. He just happened to have a photo of himself snapped at the same moment.  

Across the web, people asked, "What are the odds that in a country of one billion people, the guy standing behind you just happens to be your years-in-the-future spouse?" I told my friend (who is a talented statistician) of the pleasure I used to get demonstrating to my statistics students how commonplace rare events are. He said, “Right … you can turn [this pair of photos] around and see it as, ‘[I]n a nation with over a billion people, this is bound to have happened with someone.’” Really, the only odd thing here is that the couple happened to discover this long-ago coincidence.

I used to pose the following question to my stats classes:

Suppose you’re a telemarketer with a list of 100,000 names. You call a random name from the list and warn them that their car warranty is about to expire. Then you pick a second name from the same 100,000 names and call that person. Keep at it till you have made 150 calls. What are the chances that you will have called at least one person at least two times?

100,000 names? Only 150 calls? The odds must be infinitesimal, the students usually thought. In fact, there’s a 10% chance you’ve already hit someone twice. Make 370 calls, and the odds are over 50%. In one class, a student shouted, “Ohmigod! The same telemarketer called me twice, and I thought the odds of that must have been astronomical.” (She recognized the caller’s distinctive voice, and he remembered their earlier conversation, too.) Such coincidences happen all the time, I said. The only rarity here was that you noticed it. (A scaled-down version of this problem is called the “Birthday Problem.”)

Someone once told me of his friend, a Texan who sat on a plane next to a gabby fellow passenger. “You’re from Texas! Why, I have a friend in Texas. I wonder if you know him.” The Texan started rolling his eyes, given that his state has nearly thirty million people spread across 268,596 square miles. The gabby passenger named his friend, and it turned out to be the Texan’s next-door neighbor.

A friend of a friend had a first name-last name combination that sounded horribly obscene. The moment he arrived college, he instantly started going by a different nickname to avoid the embarrassment. Decades later, a friend of mine discussed this story (loudly, it seems) while dining out with a large group of people. One of his fellow diners said, “That’s impossible. No one would give their kid that name.” A fellow seated at the next table got up, walked over, said, “Oh, yes they would,” and flashed his driver’s license.

In one of history’s most impossible coincidences, John Adams and Thomas Jefferson both died on the 50th anniversary of the Declaration of Independence. It’s a highly unlikely event, no doubt, but it would be even more unlikely if the world had no such highly unlikely events.

This morning, someone emailed me a lovely video of a Swiss alpenhornist giving a concert performance. He explained, “Soloist is a friend of a friend. Not unusual to be two degrees separated from just about anyone in the music world. Hell, she’s probably a friend of a friend of yours too.” I looked her up on Facebook. We had no mutual Facebook friends, but two of her friends shared the same mutual friend with me. So, this musician is not a friend of friend to me (at least not on Facebook), but she is a friend of several friends of a friend. In the past I’ve come across faraway total strangers on Facebook who share multiple mutual friends with me—often from very different parts of my life. For some reason, one of my childhood accomplices from Petersburg, Virginia shows up on lots of these coincidental connections—making him the Kevin Bacon of my circles. I’ve even had strangers message me, asking how I know this guy.

Coincidental connections are everywhere. The internet just makes them easier to find. With facial recognition software crawling all over social media, I suspect that over time, miraculous discoveries like that of Xue and Ye will become commonplace and lose a bit of their magic. (Kind of a shame, really.) [After my alpenhorn discussion, I downloaded a book, The Improbability Principle: Why Coincidences Miracles, and Rare Events Happen Every Day, by statistician David J. Hand. Perhaps I’ll review it after reading it.]

So, impossible coincidences are everywhere. As the White Queen told Alice, “Why, sometimes I've believed as many as six impossible things before breakfast.”