What are the Odds?

This is Hidden Brain. I'm Shankar Vedantam. There are so many small delights that people have found themselves longing for during this past year. The sudden hush in a movie theater as the lights go down. The roar of an airplane's engines as it accelerates down a runway bound for a new destination. The happy buzz of conversation and laughter in a crowded restaurant. Another tiny pleasure that has become more rare during the pandemic The unexpected moments that we call coincidences. Staying at home has meant fewer opportunities for chance encounters, like running into a neighbor on a vacation or discovering during a chat at the water cooler that you share the same birthday as a coworker. As the pandemic is brought under control in some parts of the world and people start to move about, we are more likely to experience coincidences. And that made us think about an episode a couple of years ago when 2 different listeners called into Hidden Brain.

So I was a student at the University of Rhode Island, and we were in this writing class.

This is Amanda Birch. She told us she was chatting with the teacher of her writing class. The teacher mentioned that she lived in a small town in Vermont. The same small town, it turned out, where Amanda's mother had grown up. The teacher asked Amanda what her mother's maiden name was. Amanda told her.

And she just kind of drops her pen and she goes, you're not going to believe this, but I live in the house where your mother grew up.

The other listener who called us was Sarah Toporoff. She called us on a scratchy phone line from Paris. Sarah's an American but was living in France. One time, she was at a house party, and someone there told her there was another American in the room. So she went over to say hello.

They started talking, and so it turns out that this girl that I met at this house party in Paris grew up in the exact same house in Poughkeepsie, New York, as my father.

What are the odds? Just by chance, two listeners on different continents. calling in with such similar stories. This week on Hidden Brain, we explore the mathematics of coincidences.

That particular kind of coincidence, meeting an acquaintance or somebody you're familiar with in a very strange place, perhaps 80% of all the coincidences I've heard fall into that category.

And we ask why these little moments feel so psychologically significant.

Whenever you see something that you can't explain, the natural human instinct is to try to understand it. But that can also mean that sometimes we overexplain things.

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I work at a hotel. The blankets get washed once a week.

When I make food, I do a tribal dance in front of the microwave. I've been faking a British accent ever since I got to college 3 years ago. The messages are short, about the length of a tweet, and they range from the silly—

As a teacher, I'm not supposed to have favorite students, but I definitely do. I just want someone to look at me the way I look at tacos.

To the very serious. I'm having an abortion on Monday. My husband cheated on me. Should I forgive him? Some months ago, the folks at Whisper got in touch to tell us a story about an interesting coincidence that happened on their site. Lauren Hudson is 27, and she turned to Whisper at one of the darkest moments of her life.

I was in a really bad relationship. There was a lot of domestic violence involved, and I didn't really know what to do or how to reach out to people or how to talk about it. So I used the app to connect with other women and to kind of vent and, like, find empowerment to get away from The person that I was with.

Lauren was having trouble opening up about what was happening with her friends and family.

I think I was scared, to be completely honest. I think I had a lot of fear, and I didn't want, like, any police involved. I didn't want, like, I didn't want it to turn into, like, this whole thing.

But on Whisper, she could be anonymous. She could seek support without all the immediate consequences. It was there in one of these threads that she started talking to a man. Lauren was touched by how much compassion he seemed to have for her, a complete stranger.

It was almost as if he genuinely wanted to know how I was. Even though he didn't know who I was and he knew that it was anonymous and who knows who he's talking to, he just genuinely wanted to know if I was okay.

He was in a town nearby, another coincidence perhaps, and they started exchanging messages. They connected immediately, decided to meet in person.

We got together and we went to Applebee's and we sat and we talked for like, I think it was like almost 2 hours that we just sat and talked.

They became more and more inseparable. Lauren ended her unhealthy relationship and started dating this guy from Whisper, Eric. And everything was going great. But Lauren was a little nervous about meeting Eric's family. Eric had brought Lauren home to have dinner with them, and the whole time she was anxious.

Like, my palms were sweating and I felt like really hot, you know? So I didn't know if like if it was going down well or if I was totally blowing it. Like, I didn't know what was going on.

She was especially nervous about Eric's sister, Amanda.

You know, I knew when I first met her there the first couple times that we hung out and stuff, I knew that there was a little bit of a distance there. And I knew that every time I went over to, you know, hang out with Eric and his family, that those walls would slowly break down.

But Lauren was impatient. She and Eric were getting engaged. Lauren understood Amanda's attitude. She felt the same toward the girls her younger brother would bring home.

Maybe I'll see them next week, you know, or maybe I'll never see them again. You know what I mean? So that was kind of like what I had in my head. I was like, well, I hope that, you know, I'm thinking, I hope that they know that I'm not going anywhere.

Lauren wasn't going anywhere. She and Eric were getting engaged. Eric, in fact, had a flair for the romantic, bringing her roses, comforting her with her favorite movie after a tough day. And he would tell his family about all this. So why couldn't Lauren break through with Amanda? All this time, Lauren was still on Whisper.

Now I use it— instead of me looking for help, I use it to help others. That's my way of giving back because The Whisper community helped me when I needed them.

One day, she started exchanging messages with a woman who was going through a very painful breakup.

I sent a big message basically saying that it's okay, everyone goes through heartaches, and there will be many more people out there.

As a way to offer encouragement, Lauren told the woman about her own happy relationship. The woman asked Lauren about the nicest thing her fiancé had done for her. Lauren told her about the day Eric surprised her with her favorite movie.

On my birthday, um, I had a really bad day at work. It was like one of those days where, like, you know, you wake up and just from the moment you put your feet on the floor, like, everything goes wrong. So I was just— I didn't want to see him that day. I was just miserable. I just wanted to be home. Like, I was just like, I just want to go to bed. And he surprised me with these beautiful flowers, these roses, and he had The Notebook, which is my favorite, like, sappy love story of all time.

Lauren finished telling her story and waited for a reply.

Her first response after I sent that story to her, she's like, Lauren? Like, she— Lauren? I was like, how do you know my name? You know, because it's all anonymous. And she's like, this is Amanda.

Amanda, Eric's sister, the very person she'd been trying so desperately to connect with in real life. Lauren says this coincidence changed her whole relationship with Amanda. Connecting randomly online and revealing intimate details about themselves made the 2 women feel closer. Many of us have experienced these kinds of coincidences. You bump into your kindergarten friend your first day in college, or you meet someone at a party and discover she lives in your dad's childhood home in Poughkeepsie. When these kinds of coincidences happen in our lives, they feel like magic. But as any mathematician will tell you, things that feel unusual or even impossible are actually fairly common. Mathematician Joseph Mazur knows this, but he was reluctant to write a book that would dispel the magic.

Coincidences are wonderful stories. I don't want to blow the stories in favor of the mathematics because, you know, I was hitting a nerve on coincidences.

In the end, though, Joseph did write his book. It's called Fluke: The Math and Myth of Coincidence. It's full of stories of people who find themselves experiencing things that feel so unlikely. But Joseph Mazur says if you study this for a while, the coincidences start to fall into certain categories. And the stories we heard from our 2 listeners, as well as Lauren's story about meeting her soon-to-be sister-in-law on Whisper, are all basically the same type of coincidence. Which in itself is not a coincidence.

If you categorize these coincidences to, let's say, 10 different categories, that particular kind of coincidence— meeting an acquaintance or somebody you're familiar with in a very strange place— I would say perhaps 80% of all the coincidences I've heard fall into that category.

Joseph Mazur says the reason these coincidences feel extraordinary but actually are not so extraordinary is because of a common misconception about the number of people we think we actually know.

People think that their address book is essentially the people they know, and it turns out that any particular address book is about 1% of the people they actually know in some way. In other words, a neighbor, somebody they bump into in the street, But the address book is about 1% of the people they know.

As a result, the odds of bumping into someone you know are much greater than you think because you know many more people than you think. Sometimes math can help us understand the magic. Magic like winning the lottery, not just once, but several times. That's what happened to a woman named Joan Ginther.

In 1993, I think it was, She won $5.4 million in Texas Lotto. 13 years later, she won again, $2 million. And then a few years after that, she won $3 million. And then in 2010, she won $10 million.

What are the odds of one person winning the lottery 4 times?

Yeah, the odds are— I think somebody made a calculation, and I did too, that the odds are about 18 septillion to 1 against it happening.

18 septillion to 1. That's incomprehensible. For those who are not mathematicians, a septillion is 1 followed by 24 zeros. But Joseph Mazur says we're actually asking the wrong question. Think about it this way. If I buy a lottery ticket and I ask the question, what are the odds that I will win? My odds are very small. But if I ask, what are the odds that anyone will win the lottery in a country of hundreds of millions of people, the odds are actually much higher. So if you ask the question slightly differently, not what are the odds that Joan Ginther will win the lottery 4 times, but what are the odds that anyone will win 4 times, you get a very different answer.

I think I calculated at one point about 5 million to 1, which is not anywhere near the septillions. It's about 5 million to 1, and that takes into account the fact that we have thousands of lotteries in the world. I mean, there are many lotteries, and many of them are big-time lotteries.

So, 5 million to 1. Still unlikely, but not incomprehensible. What makes it even more comprehensible is the fact that most lottery winners don't stop gambling when they win.

And you do find that almost everybody who does win a lottery fairly big time spends all that money, or much of that money, in trying to win again.

Joseph Mazur says this might have been the case with Joan Ginther.

You win $5.4 million, you have money to play with. You have house money. So you're taking the house money to bet again. Her odds of winning a second time are better than most people's because she's got the money to play with. She wins a second time, then she's playing with more money. And you can see the, between the first winning and the second winning, it was 13 years. Between the second winning and the third, it was only 2 years. And between the third and the fourth, it was only 2 years as well.

So Joseph Mazur can wrap his head around what happened to Joan Ginther. But there are some coincidences that just defy mathematical interpretation. Joseph told me one of his favorite stories about a 19th century French poet, Émile Deschamps.

Émile Deschamps, as a teenager, he meets a man by the name of a strange name, Monsieur de Fortigou. He turns out to be an immigrant from England, and Fortigou introduces him to a plum pudding. It's a very English dish. It's almost unheard of in France.

10 years go by, and Deschamps is passing a restaurant in Paris. There's a sign on the window saying that they have plum pudding on their menu. But when Deschamps goes inside, he's told the last of the plum pudding was just sold to a gentleman sitting in the back.

The waiter calls out loud, Monsieur de Fontainebleau, would you be willing to share your plum pudding Years pass.

Deschamps is now at a dinner party with some friends. The host announces an unusual dessert will be served: plum pudding.

And Deschamps jokes that one of the guests to arrive must be Monsieur de Fortinbras. Well, soon the doorbell rings and Monsieur de Fortinbras is announced and he enters. He's an old man by now. But Deschamps recognizes him. Monsieur de Fortinbou looks around and realizes that he's in the wrong apartment. He was invited to a dinner, but not in that apartment. I love that because it's, you know, it's a triple coincidence, and it has a beautiful story element with it. I mean, that's so Magnificent. And that's why— that's the kind of coincidence I love to hear.

Joseph Mazur, professor of mathematics and author of Fluke: The Math and Myth of Coincidence. When we come back, we're going to look at what coincidences reveal about the mind. In particular, I'm fascinated by something I've observed in my own life. When I experience a coincidence, it invariably feels more meaningful to me than it does to others. Stay with us. Support for Hidden Brain comes from Instagram. Instagram Teen Accounts automatic protections limit who can contact teens, the content they can see, and the time they spend on the app. Teens are automatically placed into Teen Accounts. Teens under 16 will need a parent's permission to change any of those settings to be less strict. Learn more about teen accounts and Instagram's ongoing work to protect teens online at instagram.com/teenaccounts. Support for Hidden Brain comes from Progressive. Whoever says saving money is hard doesn't know that new customers who bundle and save with Progressive save hundreds. Because switching your home and auto insurance is better than cutting down on those iced lattes you love. Switch to Progressive and see if you could save. Get your quote today at progressive.com. Progressive Casualty Insurance Company, affiliates, and other insurers. Not available in all states. Support for Hidden Brain comes from Quince. We all have those pieces in our closet that we can count on. The versatile jacket, the perfect t-shirt. These are the pieces that make getting dressed easy. And that's what Quince does best. Quince makes elevated everyday essentials using premium materials in designs that are timeless, thoughtfully crafted, and made to be worn again and again. And when the temperatures start to drop, Quince makes it easy to build on these everyday staples with versatile layers designed to keep you warm and comfortable all season long. Find your next fall favorites at Quince. Download the Quince app for app-exclusive offers or go to quince.com. Get free shipping on your order and 365-day returns. Now available in Canada and the UK too. And when Quince asks where you heard about them, let them know it was Hidden Brain. Welcome back to Hidden Brain. I'm Shankar Vedantam. I'm fascinated by coincidences and could listen for hours to stories about them. But what is it that so fascinates us about these moments in life? At the University of Chicago, psychologist Nick Epley thinks he might have one answer. He's the author of the book Mindwise. To understand how Epley thinks about coincidences, I need to take a moment and explain his broader work to you first. Epley's book examines how we read the mental states and intentions of other people. This is a remarkably useful skill But we sometimes misapply it. We sometimes attribute intentionality to things that aren't even living, especially when they're out of the ordinary. For example, we don't personify a sunny day, but we give hurricanes a name: Katrina, Sandy, Andrew. After Hurricane Katrina, the mayor of New Orleans said, God must be mad at America. This isn't new. For centuries, people have ascribed intentionality to natural forces. We think storms, droughts, and earthquakes must have a mind of their own.

This general tendency to explain behaviors in terms of purpose or intent or, or meaning is often quite a successful thing to do. It turns out we live with other people who do have intentions and goals and foresight and planning and meaning behind their actions. And so By and large, that tendency to make those kinds of attributions to other people is a good way to learn about each other, is a good way to explain why other people did something in the past, to predict how they're likely to behave in the future. But the problem with any kind of good tool is that sometimes we use it a little too much. And that's what happens in these cases, where you're trying to explain the behavior of something that doesn't really have a sensible explanation for it. Some random event, some chance occurrence, some— physical anomaly, some weather pattern that crops up that might not in fact have a mind behind it.

This is why we're fascinated by robots that move in unpredictable ways.

What we found was that when people were given a description of an object that made it sound unpredictable, like you couldn't anticipate its actions, that's when people described it or reported that it had more Mental states. They were more willing to say that the object has a mind of its own, that it has intentions, for instance. And I don't think this is totally counterintuitive. That is, a mind— if you see, for instance, a billiard ball sitting on a table, another ball comes along and hits it, and that ball rolls off in the predicted direction, well, you don't need anything to explain that. But if you've got a ball sitting on a table and it suddenly starts rolling around randomly, Well, then you've got something to explain. And in these cases, sometimes people look to the inside of an agent, give it a mind. Maybe the ball is possessed. Maybe there's a demon controlling it, moving it around the table. Sometimes people attribute a mind to something when it's not there, but they do that as an effort to explain what's going on in this autonomous agent. And they're more likely to do that when an agent behaves somewhat unpredictably.

I was over at a friend's house. And my friend has a wonderful pool. And he had set this underwater Roomba, for the lack of a better term, to clean the bottom of the pool. So it sort of rolls around the bottom of the pool. And it goes in this completely random fashion, right?

Mm-hmm.

So it just runs for a couple of hours. It's cleaning the bottom of the pool. And I found myself fascinated just watching it because the fact that you can't predict what it's going to do next makes it just more interesting and more— Alive in a way than something that moves in a very predictable fashion.

Absolutely. Something that moves robotically, very predictably, that is just a mindless machine. But when your Roomba is going around randomly, well, now it wants to go over there a little bit. Now it wants to move over here. Now it realizes it didn't clean that side of the pool over there. Things that behave randomly start to get a little sense of a mind. Now, people aren't crazy. You don't think it's like your mother, right? Or Or your spouse who's vacuuming the living room, but it starts to look just a little bit more mindful. In fact, the Roomba is a good example. People actually do name their Roombas. You can buy outfits to dress it up. If you Google this online, you'll find all kinds of examples of people thinking of their Roomba as having perhaps a little bit more of a mind than it really does.

I want to talk to you a little bit about the subject of coincidences.

Mm-hmm.

And I thought of you and your work because it feels to me that when something unusual happens to us and we stop and we say, that is weird. I was just reading this word in a book a second ago, and someone in the room around me mentions the very same word. And it makes you stop and look up, just like I stop and look at that underwater Roomba.

Absolutely. That's a case where something unusual happens, it was unpredictable, and you try to explain it. You can explain these kinds of random coincidences in lots of different ways. There are lots of different kinds of causal structures we might be able to put on it. But some of them involve giving it more meaning than it actually has.

This is what underwater Roombas have to do with coincidences. Like a robot that does something unexpected, coincidences cry out for meaning or explanation because they're out of the ordinary. We start to seek patterns even when there aren't any.

My favorite example of this in the world of psychology is actually a phenomenon first documented by my PhD advisor Tom Gilovich and the brilliant psychologist Amos Tversky. It's the illusion of the hot hand.

In basketball—

Danny Green again from downtown! He's hit 6 in a row going back to game 2.

The myth is that when shooting free throws, basketball players get on a roll.

People believe that basketball players get a hot hand and that there are times when they're on and the chance of making a basket is higher after they've just made one than after they've just missed one.

Epley isn't saying basketball players don't have streaks. He's saying we draw the wrong conclusion about these streaks.

Let's imagine that they're a good shooter and they shoot 50% from the field. That 50% probability is going to produce a lot more clumping in the baskets than people expect. People expect a coin flip, a 50% probability, to alternate a lot more than it actually does. And so when you see randomness out in the world, it actually looks more ordered to you than it really is. 15 3-pointers for the Spurs! What a—

Epley described a party trick that demonstrates this phenomenon. You can do this too. Give everyone in the party a note card and have them write down what they think 30 coin flips would look like. H if they think it'll be heads, T if they think it'll be tails.

You then, as master magician, leave the room and you have one person come up and actually flip a coin 30 times.

That person writes down the actual sequence of coin flips on a note card and then collects all the note cards, the real and the imagined sequences, and puts them together. You, the magician, then come back into the room and identify the card that was the real coin flip.

And the way you do this is you look for the card that has the longest runs. Because people's imagination doesn't presume that a 50/50 probability will produce very long runs, and so people's imagined coin flips will alternate more than the real coin flip will, and that's how you can identify the real one from the fake ones. And I think that illusion that we have, that randomness alternates or is more chancy than it actually is, that there's less clumping, is also the thing That gives rise to these other phenomena where we see randomness in the world and we see more predictability or intentionality than actually exists in it.

When we asked our listeners to share coincidences with us, many of them wrote in with really interesting examples of things that happened to them. One thing that I was struck by is the sense that sometimes a coincidence seems more meaningful to you than it does to someone else. So for example, when I'm reading a book and I come by an unusual word, and then someone in the room mentions that same word, to me it feels like it's a sign of something. But if I were to tell you, I say, Nick, you know, this happened to me yesterday, you'd say, yeah, sure, that's going to happen once every month or so. It's exactly what you would expect if you believe the laws of probability. Why do you think it is that coincidences are more meaningful to us than to other people?

That's a really good question. I actually don't know. That I've seen any research that demonstrates that phenomena, but that strikes me as a very compelling hypothesis. I think that's probably likely to be right. And I think the reason why coincidences seem very meaningful in the first place is that you're trying to explain them. You and I both thought of the word propeller at the same time. How on earth could that be that Shankar was thinking about propeller at the same time that I was? And so you're trying to explain that. You're focusing on that event that just happened. To you. It has personal relevance to you. That makes it impactful, but it also makes you somewhat myopic. And what we don't think about are all of the other things that any of us in the room could have thought about at the same time that might also have seemed amazing to us. We're trying to explain this one thing because it's so meaningful to us. It just happened to us. Other people are less likely to be that myopic, I think, and so they would be likely to think about other things that could have happened to you, which would make it seem Less amazing to them.

We, of course, experienced this ourselves here at the show when 2 listeners called in with eerily similar coincidence stories. I described these 2 stories to Nick about women who ran into people who turned out to live in the very same house where the women's parents had grown up. So these are not just 2 coincidences. These are 2 coincidences that both happened to us at Hidden Brain, which seems Truly extraordinary, Nick.

How could that possibly be that you get these 2 that are exactly the same? It's magic is what it is, Shankar. You're magical.

That's really just all I wanted to hear you say, Nick. It's been 20 minutes to get that out of you.

Eventually, we have to give in to the laws of probability. There's nothing that can explain that, right?

Nick, I want to thank you for talking with us today.

Yeah.

Thanks for having me.

Nick Epley is the author of Mindwise How we understand what others think, believe, feel, and want. Hidden Brain is produced by Hidden Brain Media. Our production team includes Bridget McCarthy, Ryan Katz, Kristen Wong, Laura Querelle, Autumn Barnes, and Andrew Chadwick. Our executive producer is Tara Boyle. I'm Hidden Brain's executive editor. When this episode was first released back in 2016, it was the very first one in which we recognized an unsung hero. In the years since then, this moment at the end of our show has become an important part of what we do at Hidden Brain. Our very first unsung hero was Mathilde Piard, a senior project manager at NPR. She knew everything about everything. And as Hidden Brain launched into the world, she played an important role in getting it started. She was always generous with her time and generous with her knowledge. Thank you, Matthilde. If you like this episode, please consider supporting our work. You can do so by going to support.hiddenbrain.org. I'm Shankar Vedantam. See you next week.

Thank you.

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