So the average bridge player shouldn’t expect this sort of thing to happen too often in a lifetime, unless he happens to live on the planet described by evolutionary biologist Richard Dawkins in his book, The Blind Watchmaker.
If on some planet there are beings with a lifetime of a million centuries, their spotlight of comprehensible risk will extend that much farther toward the right-hand end of the continuum. They will expect to be dealt a perfect bridge hand from time to time, and will scarcely trouble to write home about it when it happens.
When Dawkins says “dealt a perfect bridge hand,” he means when someone receives a “perfect deal,” such as thirteen cards of the same suit. A “perfect hand” in bridge would be one that cannot be beaten, and involves quite different mathematical calculations. In case you are interested, the odds against being dealt a perfect hand in bridge are 169,066,442 to 1.
Anyway, even on a planet where people live for countless millennia, the prospect of all four players in a bridge game receiving perfect deals looks a little unlikely. Dawkins calculates the odds against this happening as 2,235,197,406,895,366,368,301,559,999 to 1.
As mind-boggling as those odds may seem, such an extraordinary event has, apparently, happened—at a whist club in England, back in January 1998. As reported in the paper:
Eighty-seven-year-old Hilda Golding was the first to pick up her hand. She was dealt all thirteen clubs in the pack. “I was amazed. I’d never seen anything like it before, and I’ve been playing for about forty-odd years,” she said.
Hazel Ruffles had all the diamonds. Alison Chivers held the hearts. The spades were with the dummy. Alison Chivers insists that the cards were shuffled properly. “It was an ordinary pack of cards. They were shuffled before they went on the table, and Hazel shuffled them again before they were dealt.”
The elderly members of the whist club had just beaten astronomical odds. It was actually more likely that each would win the jackpot in the national lottery and the football pools in the same week. Unfortunately it didn’t win them a penny.
So just how astonished should we be by such an event? That we know about it at all is a product of “selective reporting.” The newspapers printed the story because they had decided that it had been a remarkable thing. The perfect hand of bridge is more likely to make headlines than an imperfect one. We don’t get headlines saying “Bridge Players Are Dealt Random Shuffle of Cards.”
William Hartston, author of The Book of Numbers, believes we get too excited about coincidences. For example, he was less than impressed by the story of two golfers who hit a hole in one with successive shots. The players had the same surname but were not related. Wasn’t this rather extraordinary?
Not a bit of it, says Hartston, “First of all, let’s dispose of the little matter of the golfers having the same name. The tournament was in Wales and their shared surname was Evans.”
But Richard and Mark Evans had both hit a hole in one at the third with successive shots. What are the odds against that?
Hartston estimates that the chance of a hole in one varies from 1 in 2,780 for a top professional to 1 in about 43,000 for a club-swinging amateur. In the latter case he calculates that on any given hole, the chance of two players acing the ball with their tee shots, one after the other, would be 1.85 billion to 1.
But isn’t that pretty staggering?
Apparently not, Hartston explains, “If 2 million golfers play an average of two rounds of golf a week each, that’s more than 200 million rounds of golf a year, amounting to a total of 3.6 billion holes. That 1.85 billion to 1 shot doesn’t look so unlikely anymore, does it?” In fact, if Hartston’s calculations are correct, we should expect this sort of thing to happen somewhere about once a year.
He argues that stories and statistics like these show two things: first, that we are bad at assessing probabilities and second, that we tend to err in the direction of optimism. “Encouraged by stories of holes in one, royal flushes, and jackpot wins, we swing our golf clubs in blind hope and gamble our spare cash on impossible odds, hoping to catch the eye of Dame Fortune. Yet at the same time we play sports, where the injuries send millions to the hospital every year, we travel by car, which kills many people every day, and we smoke, which causes hundreds of thousands of deaths a year.”
In the fifty years following the first conquest of Everest by Sir Edmund Hillary and Sherpa Tenzing in May 1953, 800 people climbed the world’s highest mountain. Of those, 180 died in the attempt. William Hartston points out that the ratio of successes to deaths is roughly 5 to 1—the same odds as Russian roulette.
Accidents and misfortune, we like to think, happen to other people. Acts of incredible good fortune, we hope, will happen to us. Certainly our general inability to fully grasp the subtleties of the laws of probability can lead to some very strange attitudes toward risks in life.
A paper produced by the Said Business School points out that we all regularly run the risk of being killed in a road accident. Almost 1 man in 100 (though many fewer women) dies that way. How much, therefore, would we pay for extra safety features that would halve the risk, such as airbags and crumple zones? A thousand dollars, or perhaps as much as $2,000? But how much would you want to be paid before you would agree to cross a minefield in which there was a 1 in 100 chance of you being killed? Almost certainly more than $2,000, suggests the paper.
Anyone seriously attempting to understand the significance of coincidences (and who wants to be clearer about the relative risks in life) might find the following statistics helpful:
• The odds against winning the U.S. Powerball jackpot with one ticket: 80,089,128 to 1
• Being dealt a royal flush at poker: 649,739 to 1
• Hitting a hole in one with any one shot: 42,952 to 1
• All four players drawing perfect hands of bridge: 2,235,197,406,895,366,368,301,559,999 to 1
• Being murdered in the next year: 18,141 to 1
• Being struck by lightning: 600,000 to 1
• Dying in a railway accident: 500,000 to 1
• Dying under the wheels of a bus: 1,000,000 to 1
• Dying in a plane crash: 10,000,000 to 1
• Choking to death on food: 250,000 to 1
And the odds against two Welshmen having the same surname: 15 to 1.
What are the odds against dreams coming true? Accounts of prophetic dreams have been reported through the ages—by the ancient Assyrians and Babylonians and throughout Egyptian, Greek, and Roman civilizations. There are numerous accounts in the Bible. And they still happen.
Sharon Martens of Milwaukee, Wisconsin, was fourteen when she met and became firm friends with a boy named Michael. About a year later she had a disturbing dream—that she and Michael were at a basketball game and he told her he was leaving town the following Tuesday. Later that week, Michael approached her at school and told her his family had made the sudden decision to move to Colorado. When was he going? The following Tuesday, he told her.
Did young Sharon have some sort of psychic premonition? Or was this just coincidence? And if it was just coincidence, what would be the odds against such a thing happening? In an article published in the Washington Post in 1995, Chip Denman, a statistics lecturer at the University of Maryland, worked it out.
He made a series of complex mathematical calculations, involving various assumptions about how often we dream and the odds against any individual dream coming true. He eventually came to the conclusion that the average person, simply as the result of chance and without the help of special psychic powers, will have a dream that accurately anticipates future events, once every nineteen years. “No wonder so many of my students tell me that it has happened to them,” says Chip.
Mathematician Ian Stewart of the UK has studied the phenomenon of coincidence. He remains skeptical that the explanation for seemingly impossible chance events lies outside the realm of the laws of probability. He thinks people who assume something paranormal is going on are failing to grasp the f
acts.
Was it possible, then, to come up with a coincidence story that Ian Stewart could not explain in purely mathematical terms? Professor Stewart was prepared to rise to the challenge. Game on.
Martin Plimmer had been on vacation with his wife and children and they were playing a coin-tossing game, guessing heads or tails. His wife guessed heads or tails correctly seventeen consecutive times. Was that just coincidence?
Professor Stewart was dismissive. “Think about it mathematically. We assume heads and tails are equally likely … one half times one half seventeen times, that’s going to be about … 1 in 100,000 probability. This is fairly unusual. Something like that happened to me once. It’s just a 1 in 100,000 chance. Occasionally you get lucky.”
That was rather disappointing. Martin’s wife hadn’t, in some mysterious way, influenced the fall of the coin, or some how “read” how it had landed or attained a rare level of cosmic harmony with her children. She’d just been lucky.
For a really time-consuming holiday distraction she should have tried to flip a coin so it landed heads fifty times consecutively. Apparently to achieve this would take a million men tossing coins ten times a minute, forty hours a week—and even then it would happen only once every nine centuries. But it would happen. And the men could then, presumably, go home.
What did Ian Stewart make of the following coincidence?
At the 1997 Spanish Grand Prix, three racing drivers, Michael Schumacher, Jacques Villeneuve, and Heinz-Harold Frentzen, all lapped in exactly 1 minute 21.072 seconds. Was this not, as the astonished commentators suggested at the time, an extraordinary coincidence?
Again, Professor Stewart was not impressed. “The top drivers all lap at roughly the same speed, so it’s reasonable to assume that the three fastest times would fall inside the same tenth-of-a-second period. At intervals of a thousandth of a second, there are one hundred possible lap times for each to choose from. Assume for simplicity that each time in that range is equally likely. Then there is a 1 in 100 chance that the second driver laps in the same time as the first, and a 1 in 100 chance that the third laps in the same time as the other two—which leads to an estimate of 1 in 10,000 as the probability of the coincidence. Low enough to be striking, but not so low that we ought to feel truly amazed. It’s roughly as likely as a hole in one in golf.”
A man riding a moped in Bermuda was killed in a collision with a taxi, exactly a year after his brother had been killed—on the same street, by the same taxi driver, carrying the same passenger, and on the very same moped.
“This is another one where the chances are low, but the circumstances conspire to make it happen,” says Stewart. “The brother was using the same moped, so he obviously wasn’t superstitious. It was probably a dangerous street. The taxi driver obviously was not a good driver. This experiment is carried out millions of times every year. You don’t hear stories about someone being killed by a different taxi driver. This kind of event is very unlikely, but every so often it will happen.”
It was time for Martin Plimmer to unleash his “killer” story.
Martin had taken his six-year-old son to the doctor for a small operation. When the nurse administered an injection, Martin fainted—hitting his head as he fell. The hospital insisted he have an X ray. He arrived at the department and was told to wait. On the table in front of him was a four-year-old magazine—open at an article he had written—on the subject of headaches.
“That’s a good one.” said Professor Stewart. “It’s surprising and unusual. Things like that don’t happen to you very often, which is why we find coincidences striking. Given all the factors involved, the odds against it happening must be in the region of a 1,000,000 to 1. But how many things happen to you in a day? A thousand things? At least. Over three years … a thousand days of a thousand things a day, a million things happen to you. In among those there will be one whose chances are one in a million. So about once every three years something like that ought to happen to you. If it’s happening to you more often than that, then it is getting interesting mathematically.”
Professor Stewart says the reason we tend to be so amazed when these coincidences occur is not simply because they occur—but because they happen to us. “Of all the people in the world it could have happened to, it happened to you. The Universe picked you. And there’s no explanation for that.”
He adds that our intuition is worse than useless when we think about coincidences. “We’re amazed when we bump into friends in unusual places, because we expect random events to be evenly distributed—so statistical clumps surprise us. We think that a ‘typical’ draw in a lottery is something like 5, 14, 27, 36, 39, 45—but that 1, 2, 3, 19, 20, 21 is far less likely. In fact these two sets of numbers have the same probability—1 in 13,983,815. Sequences of six random numbers are more likely to be clumpy than not.”
What then did Professor Stewart make of one of the most famous of all coincidence stories—that which connects the lives and deaths of Presidents Abraham Lincoln and John F. Kennedy?
Abraham Lincoln was elected to Congress in 1846. Kennedy was elected in 1946. Lincoln was elected president in 1860, Kennedy in 1960. Their surnames each contain seven letters. Both were concerned with civil rights. Both were shot on a Friday. Both were shot in the head. Both were assassinated by men with three names comprising fifteen letters. John Wilkes Booth who assassinated Lincoln was born in 1839. Lee Harvey Oswald who assassinated Kennedy was born in 1939. It goes on and on.…
“If you just take the list of things, it sounds like a very unlikely chain of events. But it’s numerology. People are looking for the things that are the same and ignoring all the things that are different. You focus on the fact that some names have the same number of letters but other names don’t. How many letters on average would three names have? Well fifteen is probably close to average. The fact that they were born one hundred years apart means their careers are likely to track each other roughly one hundred years apart. Both were shot on a Friday—well there is a 1 in 7 chance.
“If you play these games and look for similarities and are prepared to be imaginative about what you look for and only count the things that are similar, I suspect you can take any two people on the planet and find an amazing amount of things in common.
“The fact is that they are both human beings, which means they have a lot in common to start with. You just have to find out what it is.”
And there is evidence to back him up on that. Awhile back The Skeptical Inquirer held a competition to find “amazing coincidences” between other world leaders. The winning entry unearthed sixteen uncanny similarities between Kennedy and President Alvaro Obregon of Mexico.
Arthur Koestler suggested that a possible explanation for coincidences is that like things in the universe may be attracted to each other. Did Ian Stewart have any sympathy with that view?
“There is a sense in which that is true,” said Professor Stewart. “But for obvious reasons. People who travel by aircraft a lot will be attracted to one another in airports. It’s not a surprise if lots of coincidences happen to me in airports, because I spend a lot of time in them.
“As regards a more mystic kind of attraction of like things, I’m not convinced. Some people suggest that there is a secret hidden order to the universe and it is our job as scientists to work it out. But that kind of unity in the universe is on such a deep level—of fundamental particles all obeying the same rules—that it does not translate into anything meaningful on the level of people in terms of an obvious association of like with like.…”
And then suddenly, and unexpectedly, a chink appeared in the mathematician’s armored skepticism about the existence of some sort of synchronistic force that creates coincidences.
“… but on the other hand I wouldn’t say it was nonsense. I mean the universe is a very strange place and it does function in ways we don’t understand very well.”
Did this mean that it was, after all, possible to think of a coincidence sc
enario that Professor Stewart could not dismiss as the result of pure chance—that was “beyond coincidence”?
“The place where I lose confidence in my explanations,” he said, “is when I get to the point when I’m not explaining—but explaining away.”
What, for example, if we were talking about meteorites and one landed on a nearby building? Could he explain that away?
“I don’t think I could. It would be very difficult. At the very least it would be a remarkable thing—an astonishing thing.”
And what about the chances of actually being struck by a meteorite?
The chances, he reckoned, were astronomically slim. The only known instance was in 1954 when a nine-pound meteorite crashed through the roof of Ann Hodges’s home in Sylacauga, Alabama, and struck her in the hip as she slept on a couch. She escaped with a large bruise.
So did Ian Stewart think it was safe to leave the building?
“Well, the problem is you just don’t know. You could go somewhere else and that turns out to be the place that gets hit.”
You’ll never guess what happened when we walked out of the building …
Nothing.
Two
COINCIDENCE ON THE RAMPAGE
1
IT’S A SMALL WORLD
When coincidence taps us on the shoulder in the form of an old friend in a strange place, we marvel at what a small world we live in.
Everyone agrees that the invention of the airplane has made the world an even smaller place. Not so small you could put it in your pocket, perhaps, but small enough to travel halfway around the world in the time it takes to watch half a dozen bad movies.
In fact the world has remained resolutely the same size (7,925 miles in diameter at the equator the last anyone checked), give or take a few quarters of an inch for natural shrinkage. And that’s pretty big really, though not as big as Jupiter, of course, which is one thousand times greater in volume. If coincidences occur in direct proportion to the smallness of the planet, then presumably they occur one thousand times less frequently on Jupiter. Someone should look into that.
Beyond Coincidence Page 11