We are usually surprised when a coincidence happens. It often amuses or delights us. Perhaps we bump into that friend we haven’t seen for a while, or we hear the same name or phrase twice in one day, or we win a small amount on the lottery.
Most coincidences are small. Occasionally they can be unexpected and very dramatic. Often they can have far-reaching consequences that persist over generations, for example I certainly wouldn’t be writing this if my parents hadn’t been accidentally introduced to each other all those years ago.
Coincidences can also be unwelcome. Who hasn’t been the victim of an accident, which on another day under similar circumstances, wouldn’t have occurred? We often call it being in the wrong place at the wrong time.
Superstitious people tend to look at coincidences as signs from God, or some other hidden force. They wonder if they’ve been singled out for some reason. Superstitions like this are nonsense of course. Coincidences are all about mathematical probability.
Should we be surprised by coincidence?
When I was in Secondary School, I was amazed when the Maths teacher declared one day “I bet that two children in this class have the same birthday”.
We looked at him as if he was some sort of lunatic. Out of our class of only twenty-four children, what were the chances that two shared the same birthday? A quick mental calculation told me that there would be an average of two birthdays per month in the class. For any given month, what are the chances of those being on the same day? Just about zero, I thought. If the teacher had actually been taking bets I would have been first in the queue, offering my tuck shop money in return for a betting slip.
You can thus imagine everyone’s surprise when, after a straw poll of the class, we found out Simkins (minor) and Spotty Trevethick were both born on the same day in July. How could this be? It seemed to go against all reason. I struggled to make sense of it. After all there were only twenty-four people in the class. The teacher didn’t help by standing there with a smug grin on his face, but at least I had the consolation of retaining my tuck shop money.
It turns out that the teacher was making a reasonable bet. For a class of twenty-four children, the odds are better than fifty-fifty that two kids in the class will have the same birthday.
If the class size is fifty, there’s over a ninety-seven percent chance that two people will have the same birthday. In other words it’s almost certain.
How can this be? How can something which feels so improbable actually be rather probable?
It all comes down to mathematics. For the reader this will probably be either very good news or very bad news. For the arithmophobes amongst you I have spared you the proof of the classroom birthdays at this juncture, but for the bi(nomial)-curious I’ve added a section at the end.
Mathematics…. that intangible thing which can be described by one person as cold, hard and boring; by another as the warmest, most beautiful expression of the Universe itself.
But I digress. Let’s get back to coincidences and try to keep a human face to it.
Let me regale you with three further coincidences which I know to be true. The first happened to me. The other two happened to friends of mine, and just as certainly I know them to be true.
Coincidence number one: The young couple
It was 1983 and I was living in Liverpool. A frequent companion in these times was a Welshman named Dai. He and I would philosophise over a pint (or two). The more we drank, the more we understood. We enjoyed the City. The people. The times. The music. The culture.
But then a strange thing started to happen. No matter where we were, Dai and I would notice a couple enter our bar. They stood out. He was tall. She was small. A good-looking couple and apparently inseparable.
It was a joke at first. “Oh look, there’s that couple again” we would say, but after a few far-fetched coincidences we started to wonder. Were they following us? Were we under some sort of surveillance? Why would they be doing this?
We eventually reasoned it was just coincidence. Besides, by now they had noticed us too. In particular she would catch my eye. I was usually embarrassed. Perhaps they thought we were stalking them.
1983 disappeared and Dai and I went our separate ways. The coincidence of the young couple was soon forgotten. Dai and I pretty much lost contact over the next three years. It was harder to stay in touch in the days before the ubiquitous Internet.
Then one day, out of the blue, Dai telephoned to invite me down to his house for the weekend. I readily agreed. It was by now three years since I’d seen him. It would be good to share another pint (or two) and see how our philosophies had changed.
I arrived at Epping, two hundred miles from Liverpool, on a cold Friday night. Christmas approached and a frosty magical feeling was in the air.
“How about a quiet inn, right on the edge of the forest?” suggested Dai. “It’s my regular watering hole now.”
“Perfect,” I replied, and we made haste, anxious for some cosy cheer against the winter night.
The country inn glowed in the night. It welcomed us with uneven floors, a beery smell and a crackling fire. We settled down in a quiet corner, intent on covering the past few years over the next few hours. There wasn’t a better place on Earth to do it.
One pint slipped away. Then another and another. Two old friends with much to share.
Our reminisces were rudely interrupted, however, by the old wooden door of the inn. After some rattling of the latch the door swung wide open. In swept a couple, accompanied in their gait by a flood of icy air. Yes, it was the same couple from all those years ago in Liverpool. I looked at Dai and he looked at me. We both slipped into shock and couldn’t speak. The young girl glanced over in our general direction. She looked at me and I looked at her, and a thousand thoughts passed between us in a second.
I am ashamed to say that we didn’t speak to the young couple that night. It is one of my regrets. Perhaps it was the shock. If I ever see them again I promise you I shall be better prepared.
Coincidence number two: Extra homework
One summer, a teacher friend of mine took a hiking trip of a lifetime to explore the foothills of North-East Pakistan.
She spent many days hiking with close freinds, taking the time to explore the beautiful countryside. They met some wonderful and very hospitable people.
On one occasion, after a particularly tiresome trek through some sparse countryside, they happened across a small village. The locals all rushed out at the rarity of such visitors and made my friend and her party very welcome.
Imagine my friends surprise when she recognised one of the villagers. It was one of her own pupils from her school back in England. She had been teaching her a few weeks previous!
It turns out that the pupil was visiting relations in the village with her own family.
The best bit of this story, after the initial shock, was that the young girl thought that her teacher had come all that way to bring her some extra homework!
Coincidence number three: Hiding in the attic
Forty years ago, when a work colleague of mine (Brian) was a small boy in Devon, property developers started to build new houses on the land next to his parent’s home.
The bricklayers, carpenters, plumbers and plasterers would work by day. Brian and two young friends would play by night. The houses were their castles, their forts, their hide-outs. As the houses grew, so did the boys adventures.
One day, Brian and his friends were innocently playing upstairs in a house, when they heard footsteps on the bare wooden floors below. It was the new owners making an unexpected and out-of-hours inspection of their new property.
Brian and his friends panicked. They must not be caught. In their young imaginations they reasoned they would certainly be captured and imprisoned, maybe even shot. Brian took the lead. He quickly and heroically ushered his friends up a wooden ladder into the attic, bravely bringing up the rear himself. Once inside the confined, dark space, the three boys inched gingerly along the rafters until they could just about squeeze behind a thick brick chimney-stack.
The boys heard the footsteps from below get louder as they echoed on the bare wooden stairs. The boys shook when a stern male voice called out: “Anyone there? Show yourselves!”
Brian and his friends held their breath and tried their hardest not to move. The footsteps searched the bedrooms before eventually starting on the attic ladder itself. All three boys trembled against each other as a man hauled himself into the attic and shouted out again “IS ANYONE THERE?”
After what seems like a year to the boys, but in reality was probably about 20 seconds, the man climbed back down the ladder and engaged in conversation with a woman below. It wasn’t long before the boys were truly alone, and they were able to make their escape, never to return.
Thirty years later, Brian and his wife are in a small hotel in the Dordogne region of France, chatting over breakfast with an elderly English couple they’d met that morning.
The conversation moved around to Devon, where the couple used to live and Brian grew up. By coincidence they realised they used to live in the same village. It’s not long before the English couple told them about their haunted house. About how, when it had just been built, they had both heard the sound of children playing upstairs, but on close and extensive examination there was nobody there. It couldn’t have been real children because there was no way for them to get out.
It quickly became apparent that the elderly couple were the people who had interrupted Brian and his friends thirty years earlier.
Brian, like me with the young couple in the earlier tale, was somehow frozen by this amazing coincidence and dared not speak and declare himself.
I personally think he was worried he was going to be shot.
These three coincidences all concern the meeting up of people who have been separated by some distance and for some time. As human beings these often rank amongst the rarest of coincidences which happen to us.
But in the same way that you occasionally share a birthday with a classmate or a work colleague, these rarer coincidences also have a mathematical probability and a finite chance of happening. It’s just that, in most cases, the probability is so small that they may only happen once or twice in a lifetime.
Do coincidences happen in nature? Of course they do; they’re everywhere. There’s millions of coincidences happening all the time, every second of the day.
The one that never fails to amaze me is a solar eclipse viewed from Earth.
“Why is this a coincidence?” I hear you ask.
Anyone who has experienced a full solar eclipse will tell you how magical it is. The moon slowing inching in front of the sun until, for a few wonderful seconds, the moon completely obscures the bright star behind it. Because the objects are the same size, wonderful corona effects are often seen around the rim of the black moon.
If the moon was any smaller, or the Earth a bit closer to the sun, we would never see a total eclipse. The moon would just be a smaller dark circle in front of the sun. Conversely if the moon was any larger, or the Earth a bit further away from the sun, the moon would completely obliterate the sun’s image, with none of the corona effects.
The coincidence is that, at this particular moment in our Solar System’s history, the sun’s diameter is about 400 times larger than that of the moon and the sun is also about 400 times farther away. So the sun and moon appear to be the same size as seen from Earth. Now that is a very, very slim chance.
In nature, coincidences often have billions of years over which to operate, including the wonderful coincidence of life itself. A remarkable event only needs a fraction of a second in a few billion years, but that is perhaps the subject of another blog.
So where is this all going?
What I’m trying to point out that coincidences are not all that rare. In fact they are the norm rather than the exception.
If we think back to the birthday example, the chance of me having the same birthday as someone else in the class is quite small. But two people in the class – Simkins (minor) and Spotty Trevethick – experienced quite a remarkable coincidence with no effort at all.
The chances of me winning a lot of money on the lottery are so small that I can confidently predict that I will never, ever win it in my lifetime. Even if I bought a thousand tickets a week. But at least one person will win a fortune tomorrow.
Coincidences are common when you consider all the different events that can happen to all the different things in the world, or indeed the Universe, over time. Birthdays and the Lottery are just very, very narrow examples of one thing.
If you pick one particular thing from the myriad bits of information we typically encounter during a day, the chances are quite small that there will be a coincidental occurrence involving that one thing, but if you take everything that happens in day, a week, a month or a lifetime, there is a strong probability that there will be a coincidence with something.
My conclusion is simple: We should never be surprised by a coincidence. If you have a day where you don’t have one, I strongly urge you to get out more.
Brief mathematical explanation of the class birthday coincidence.
**** Arithmophobes: Warning! Numbers ahead! ****
I’ll start gently: Most people look at the birthday situation from a personal perspective. We say “What is the chance that someone else in the class has the same birthday as me?”
The answer is actually quite low (intuitively and factually) and this is what fools us. If I share the class with one other person, the probability of them NOT having my birthday is 364/365, i.e. they can have any birthday but mine. Note: with this type of problem it’s easier to work out the probability of something NOT happening, and from this derive the probability of something happening, remembering that in probability theory:
Probability (event happens) + Probability (event doesn’t happen) = 1.
Phew, I hope you’re still with me, but feel free to bale out now. I’m also trying to keep things simple by ignoring leap years, and effects such as some months being more popular than others for births.
If we now add another person into the class, the probability of the new person NOT having my birthday is also 364/365. To get the probability of NEITHER of these two people having the same birthday as me, we have to multiply the probabilities together, i.e. (364/365) raised to the power of 2. As each new person joins the class we multiply them in again, until with a class of 24 we get the probability of nobody having my birthday as being (364/365) raised to the power of 23, or about 0.9388. Therefore the probability of someone having my birthday is about 0.0612, or a little over 6%.
This isn’t very likely, as we suspected, but with the class birthday scenario what we actually need to do is to look at it from everybody else’s point of view, as well as our own, all at the same time.
We’ve just seen that if there were only two people in the class, for example Simkins and I, the chances of us having the same birthday is small. If we now introduce Trevethick to the class, then there is the additional probability that I share a birthday with him, but there is the further probability that Simkins and Trevethick share the same birthday (this is quite independent of me).
And so it goes on as more people are added. We have to compare every new person against everyone else, not just ourselves, until we get to our class of twenty four people, where the maths shows us that it is actually more likely (than not) that two people will share the same birthday.
The equation for this – the probability that no two people share the same birthday in a class of 24 – involves a minor modification to the previous equation, as follows:
((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-23)/365)
This yields an answer of about 0.46, which means there is about a 54% chance of two people in a class of 24 sharing the same birthday, i.e. better than 50-50.
If anybody has been checking my maths, can I (again) suggest that you get out more…