March 25, 2024
Gresham College

The Mathematics of Coincidence - Sarah Hart

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Gresham College Lectures

The Mathematics of Coincidence - Sarah Hart

Mar 25, 2024

Gresham College

We regularly hear of amazing coincidences – people winning the lottery twice, or getting a phone call from a long-lost friend just when you were thinking about them. Is this telepathy? Is there a greater power at work when someone survives seven lightning strikes?

There can be terrible consequences from the misunderstanding of coincidence.

This lecture was recorded by Sarah Hart on 5th March 2024 at Barnard's Inn Hall, London

The transcript and downloadable versions of the lecture are available from the Gresham College website:

https://www.gresham.ac.uk/watch-now/maths-coincidence

Gresham College has offered free public lectures for over 400 years, thanks to the generosity of our supporters. There are currently over 2,500 lectures free to access. We believe that everyone should have the opportunity to learn from some of the greatest minds. To support Gresham's mission, please consider making a donation: https://gresham.ac.uk/support/

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We regularly hear of amazing coincidences – people winning the lottery twice, or getting a phone call from a long-lost friend just when you were thinking about them. Is this telepathy? Is there a greater power at work when someone survives seven lightning strikes?

There can be terrible consequences from the misunderstanding of coincidence.

This lecture was recorded by Sarah Hart on 5th March 2024 at Barnard's Inn Hall, London

The transcript and downloadable versions of the lecture are available from the Gresham College website:

https://www.gresham.ac.uk/watch-now/maths-coincidence

Gresham College has offered free public lectures for over 400 years, thanks to the generosity of our supporters. There are currently over 2,500 lectures free to access. We believe that everyone should have the opportunity to learn from some of the greatest minds. To support Gresham's mission, please consider making a donation: https://gresham.ac.uk/support/

Website: https://gresham.ac.uk

Twitter: https://twitter.com/greshamcollege

Facebook: https://facebook.com/greshamcollege

Instagram: https://instagram.com/greshamcollege

Is it more than chance when one person survives multiple lightning strikes and another wins the lottery jackpot? Not once, but twice. Mathematics can give us the tools to decide when events really are coincidences and when other factors might be at play. Getting this right has implications everywhere from the science lab to the law courts. But let's begin with someone who you could think of as being either very lucky or very unlucky. The Gresham College lecture that you're listening to right now is giving you knowledge and insight from one of the world's leading academic experts, making it takes a lot of time, but because we want to encourage a love of learning, we think it's well worth it. We never make you pay for lectures, although donations are needed. All we ask in return is this. Send a link to this lecture to someone you think would benefit. And if you haven't already, click the follow or subscribe button from wherever you are listening right now. Now, let's get back to the lecture. Meet Roy Sullivan, and he's bending down to show us the top of his hat, which has scorch marks in it from where he was struck by lightning, but he survived. So lucky or unlucky, you decide if we're gonna really understand what the chances of this kind of event happening and whether we should attribute it to some kind of divine intervention or not. We need to think a bit about probability. And I want to start by giving you an example of a coincidence that happened to me. Not as unlikely as a lightning strike, but I mention it because just on this one occasion, I can tell you the exact probability of it happening. So this is a couple of years ago, my daughter and I, we went to visit York, lovely city, and we got hungry. We thought, we'll stop in a cafe, have something to eat, and we'll go to the lou while we're there. I went up to buy the coffee. Um, as I was paying, you know, putting my pin into the machine, I said, could we use the facilities? Uh, the lady said yes. And so she handed me a piece of paper, which I thought was my receipt, but it had my pin number written on it. So, oh, no, by the way, just for the avoidance of doubt, not my opinion. Different number. So anyway, I, I was, oh, startled then I realized, no, this is the key code to use the facilities, but it was just for a moment, I was a little bit freaked out, I have to admit. So we can work out the exact probability of this, because here is a four digit number which matched the four digit number. I knew I had just typed into the, the card reader. So how many four digit numbers are there? What are the chances of two of them matching? Well, they've got their key card and I've got my pin. So if my pin's gonna match, it's gotta match the first number. That's a one in 10 chance. And the second number one in 10, and the third and the fourth. So the probability of that happening, uh, is one in 10,000, 10 times 10 times 10 times 10, okay? So that's the chance of that. So it's pretty unlikely. And, uh, one side, you know, recovered from that mild surprise, I could, uh, drink my coffee as normal <laugh>. So we know the exact probability of that. That's quite unlikely. But actually, for, for it to happen to me, yes, unlikely a surprise. But there are lots of tourists in New York. The year that we went, I looked it up, 8.9 million people visited that beautiful city and each of them are likely gonna have one, probably several cards, bank cards with pins. Uh, so I was thinking, well, can we be sure? How do we know how many? And I thought, well, adults normally have at least two of these. You probably have a debit card and a credit card, maybe more. And of course, newborn babies don't have any. But let's say that on average, you know, there's one purpose. So there are 8.9 million pins wandering around New York at some point in the year. Um, so if it's one in 10,000 chance and they are kind of randomly distributed, which pretty much they are, we would expect 890 people in York during the year to have the same, uh, pin as the key code in that cafe, right? 890, that's nearly three a day. So I don't feel so special anymore. And that's the first thing we have to observe about probability. But even if it's very unlikely, it's probably gonna happen to someone if you have enough people involved. So that's sort of a bit of a paradox, but it's not really that even very unlikely events are likely to occur in a big enough population if you have enough opportunities for them to occur. So let's get back to poor old slash lucky old Roy Sullivan. Now, we would need to know if we're gonna understand just how unlikely this is, we'd need to know the probability of being struck by lightning. Happily, the US weather services work this out for us. And they have an estimate that the, uh, probability of being struck by lightning at least once in your life, at least in in the US probably in other places, two, is one in 15,300. So it's a bit less likely than my very exciting cafe coincidence. But, you know, it's not incredibly, incredibly unlikely. However, this guy is struck multiple times, seven times. In fact, he was struck by lightning during his life and survived every single time. So, so you could think about how can we work out the chance of being struck twice by lightning in your life? Now, when we calculate probabilities with things like tossing coins or rolling dice, we might say, okay, if I, if I toss a coin, the chance of it coming up heads is, is a half 50 50. And if I toss two coins, then the chance of both of them coming up heads is a half times a half, right? One quarter. So you might say, well, okay, the chance then of being struck by lightning twice would be one in 15,000 times, one in 15,000, one in 243 ish, 234 ish million. But that is not the correct calculation. And the reason is we may have noticed human beings are not coins, <laugh>. Um, those, the, the chance of us being struck by lightning is gonna vary from person to person based on different factors. And so being struck once by lightning is not completely independent event from being struck again by lightning because maybe you are the sort of person who's likely to get struck by lightning. What do I mean by that? Not that you are cursed by the gods, hopefully, but so many factors can intervene your geographical location. So in, in America, uh, states that are on the, the east coast, maybe towards the south, they have higher rainfall, they have more thunderstorms, and therefore more lightning than other places. So geographical location matters for you. Um, I, I won't speculate on the causes for the fact that of all the people struck by lightning, 84% are men and only 16% are women. Um, I don't know, let's not even speculate on my, that might be another factor though, is the job that you do. Uh, if, if you are working indoors a lot, you're gonna have be less, less likely to be struck by lightning. So if we gather those kind of factors together, we could say, you know, this is an average likelihood, but people will have different personal likelihoods, um, in order to have to have be struck by lightning. So if you are, I dunno, a woman working in a library in Alaska, you are much less likely to be struck by lightning van, a man working as a park ranger in Virginia. So no guesses what Roy Sullivan's job was. Yes, he was a park ranger in Virginia. So he's much, much, much more likely than this overall, overall average to get struck by lightning. I mean, of course it's still really unlikely to be struck seven times. And that's why, you know, we know about him. That's why he's, he's an interesting example. But we have to be careful about using average, uh, likelihoods in those kind of situations. He, of all the people in America, was really high ranking in his chances to be someone who would be struck multiple times. But if we think about very unlikely events, sometimes it seems so improbable that, that they could happen by chance, that we start to wonder is something else going on here? And sometimes that happens when there are people who appear to be making amazing predictions of the future. Premonitions clairvoyance, for instance, uh, there was a woman called Jean Dixon who was, you know, proclaimed herself to be a psychic. Now she predicted that the 1960 US election would be won by a Democrat. Okay, that's not so impressive. But then who would be assassinated in office? And for those of you who are history buffs, you may have heard of, uh, John F. Kennedy. And indeed that's exactly what happened. So this seems really kind of spooky thing to predict. Very, very unlikely that you could predict that by chance. So surely there's something more going on. Well, let's discuss this and lead into this with thinking about really rare, unlikely seeming events, the one in a million kind of event. Can we ever expect to see things if something that's as unlikely as this happens, do we, and should we conclude that something more than chance is at play? Well, let's, again, let's have a little warmup for this. Uh, these are big numbers to work with. Let's think about the kind of event maybe that that would be not dependent on your geo geographical location or something like that. Let's imagine that all of us tonight, you know, we go to bed and in the morning we wake up and we write down something that happened in our dream. And then we see if anyone has predicted the future. Did any of those dreams actually come true in real life?'cause occasionally we hear about things like this, very unlikely to, to happen. So instead of dreams, let's think about a, a much more likely event. Let's say we're all rolling dice. Now there are some dice. So if I roll one die and I'm trying to get a six, the chances of me doing it is one in six, about 17%. But let's say the group of us are all try, we all are rolling a die, die together, uh, one each. And we are trying to say, can the group collectively get a six, at least one six? Well then things get better for us. So if two of us, if there are two of us now each roll a die, um, then, and we are hoping to get some, one of us will get a six, at least one of us. So there's different ways that could happen. I could get one or you could get one or we could both get one. So instead of working all those out separately, it's a much quicker way of doing that calculation. And it's to notice that the probability of rolling at least one six, um, is one minus the probability of not rolling any sixes at all. So if the probability of rolling a six with, with one dye is one six, then the probability of not doing that is five six. So if both the dice do not roll a six, and they are, that's it, those independent events. So we can multiply the probabilities and we see that the probability of rolling no sixes at all is five six times five six. And so if we subtract that from one, we get the probability of rolling at least one, six, maybe both of us get lucky, but at least one. And we see that it's actually about 31%. So we've nearly doubled our chances. And if we have three dice and we want to roll at least one six, same idea, and you now it's a 42% chance that at least one of us will will get a six. And if you have four dice or four people rolling a dye, then it goes up to above 50%. So we could more likely than not that someone's gonna roll a six. Now of course, it's easier to role a six than it is to have a preliminary dream that comes true. But the same principle is going to apply. If you have a group of people, um, all kind of writing down their dreams or making a prediction, then we can look at the chances that at least one of those predictions will be right in exactly the same way, just bigger numbers. So let's, we might think about a one in a million kind of event, but just for the moment, let's use our favorite number. Think about the one in n event. N is the mathematician's favorite number. So let's imagine n is a very big, big number, right? So the probability of some event having a predictive dream or something else is one in n let's say. Then if we have a bunch of people who all have an opportunity to, to have this occur, like predicting the future in a dream, then the chance that none of them will experience that is we've got one minus one over n. That's the chance of one person not experiencing that unlikely event. Uh, but there are K people altogether. So we raise that to the power K, that is the chance of none of those K people experiencing that event. Alright? Now there's a really cute rule of thumb I'm gonna show you now, if you've got this event that's unlikely to happen, a one in end chance of it happening, if we have end people and look at what happens to that expression as end changes, a really nice thing happens. I'm gonna show you the graph. This is the graph of that strange looking function, one minus one over N race power n Um, as n gets bigger, I've put it on a logarithmic scale. So you can see n getting bigger on the, on the X axis. And then we are plotting this value. And you can see that it very quickly approaches some constant value. It's never quite reaches it, but it getting really, really close. As soon as n is big enough, and even n you know, around a hundred bigger than that, we're getting closer and closer to this number, which is sort of about 0.37 somewhere around there. So if, if you are a mathematician, you might recognize what this number is. Uh, well it's one over EE the famous constant. The natural algorithm doesn't matter if you don't know what that number is, but this is one over it <laugh> and it's roughly 0.3, six, seven and some other stuff. So as n gets bigger, and for the purposes of this talk, uh, a hundred equals infinity, then we can make this approximation that, that we get this number 0.367. So that means that about 37% of the time in a group of n people all having the opportunity for this one in n type of event, 37% of the time nobody will experience it. But that means that 63% of the time, at least one person will experience this rare event. So if we have a million people, um, who in the morning write down their dream or write down a prediction, 63% of the time, if it is a one in a million shot, um, at least one person will experience it. And of course those are the people that phone up the radio station and say, I can predict the future. So that's, uh, kind of an interesting thing to observe. Again, it's if you've got enough people, it's gonna happen. Now, this is, uh, describing a situation where you've got a big bunch of people all having one opportunity for a rare event to happen. But we can turn it the other way round and think about one person having many opportunities for an event to happen. And exactly the same mathematics tells us, if you've got this very rare event, a one in a million or one in end type of event, um, if you have n opportunities for it to occur, then again there's a 63% chance that it'll happen at least once. And okay, what about if you wanna know when it becomes more likely than not? Well, you have to do a bit more calculated. But it turns out, um, if you have at least this many 0.6, nine, four ish, uh, n attempts at this thing or opportunities for this thing to happen, then it becomes more likely than not that it will happen at least once. So if you've got a one in a million shot, incredibly rare freak, spooky coincidence, I mean choose your favorite thing. You know, you are, you are just thinking about someone you haven't seen for 20 years, the phone rings and it's them. Or you go into a random secondhand bookshop when you're on your, on holiday somewhere and you pick up a book and you find it used to belong to your father or, you know, name your freaky coincidence, A one in a million shot. We have a few of these, you know, many of these potentially can happen to us in the course of a day if we have. So n is a million right now. So that means if we have more than 694,000 opportunities for such a thing to occur, then it's more likely than not that, that it will occur. So let's think about how many opportunities one would have in one's life for a really rare event like that to happen. So during the course of our day, lots of chances, you know, you're walking around, you could bump into people who you haven't seen for 40 years, or you could get that phone call outta the blue when you had a dream about that person. Or you could, yes, at the night in nighttime, you could have this preliminary dream, all sorts of things. So I'm gonna guesstimate one opportunity per hour on average for a one in a million kind of event to happen. A really freaky coincidence. So how many hours are we going to be alive for not starting from now over the course of our life? Well, based on the UK life expectancy, it's about, uh, 720,000. So have I fiddled the figures a bit? Maybe, well, you can decide <laugh>, but look, that's more than 694,000. So I, I suggest that we all have a better than average chart, better than even chance of a one in a million event at some time in our lives. So maybe these things, you know, we, we, of course they're surprising and exciting when they happen, but they are going to happen. Now, the amazing psychic clairvoyant, Jean Dixon had something else on her side. Uh, and I'm gonna you three rules for how to become very successful. Clairvoyant psychic. Uh, so this is her famous prediction that, that the Kennedy assassination, but here are some of her other predictions. Um, she also predicted that <laugh> the Now and she changed her mind and said, actually Nixon's gonna win. Um, she said that Russia would be the first country to put a man on the moon. She said that World War II would be begin in 1958 and where we'll have to see <laugh> if, if China actually does invade Russia in 2025. I hope not. But look at this list of predictions. We, uh, remember, remember we hear about the first one 'cause it happened. But there are two rules, and I'll give you the third and a second to, to improve your like, ability to boast about successful predictions. Make a lot of predictions, and then forget the ones you got wrong. So if you are absent-minded fortune teller who's very busy, you're gonna, you're gonna be able to say, oh, I was right about that. And just forget about all the others. Now Nostradamus here can show us the third rule of successful fortune telling. I'll give you an example of a genuine, uh, prediction that he made. And it's this in the world, there'll be made a king who will have little peace and a short life. So correct, that's definitely happened many times since Nostradamus, but that's so vague, you know, there's nothing we can do with it. So if you are very vague, then you can retrospectively say, oh, well I, that's what I meant when I said that. You know, there'll be a big apple in in the west. I meant that apple computers would <laugh>. What? So, okay, vague, many predictions be forgetful. That's how to be a successful fortune teller, um, without having to rely on even these coincidences that we know will happen in our lifetimes. So, okay, she got lucky with that, uh, <laugh> prediction that she made Jean Dixon. But there are sometimes examples of what seem incredible luck and can we really explain those away? Well, let's see how we get on. Here are two very, very lucky people. Uh, even Marie Adams won the New Jersey lottery, the jackpot in the New Jersey lottery twice in the space of a year. No one's done that in the UK yet, at the time of speaking right now could happen any moment. But this, uh, chap Mike McDermott, he won the next best thing. He twice managed in, I bet he was obscurely disappointed, right?<laugh>, he, he twice match five balls in the bonus ball, which is like the next sort of the second prize level in the national lottery in the uk uh, in the space of a year. So these, I mean the odds of that happening are, are tiny. Uh, it's really, really unlikely for it to happen. However, <laugh>, however, we can again, think about the fact that although of course it's extraordinarily unlikely that that particular person will win the lottery twice. There are lots of people who win the lottery in, in the world every year. So again, we do a little bit of a back of the envelope calculation. Let's think about, so every country, pretty much every country has a national lottery and many countries have lots and lots of lotteries. Uh, because in America, for example, every state will have a lottery. So that's another 50. So conservative estimate, let's say there are 200 national or state lotteries around the world, and each one, um, will have on average two draws every week. So in Britain we have two draws every week. Some places they have three. Um, but let's say a hundred draws each year in each of 200 lotteries. That means 20,000 lottery draws around the world each year, each of of which can give us a winner. Now, some don't give us any winners, you know, then there's a rollover. Uh, but others, in other weeks, several people should have to share the jackpot.'cause there are several winning tickets. So again, let's sort of average it out and say we might expect one winner for each of these 20,000 draws in the year. So that gives us 20,000 jackpot winners each year being created. Now, it's a weird thing of course to win the lottery twice, you have to enter it twice. It does seem to be the case that even people who win the jackpot, they carry on buying lottery tickets. You think, well, aren't they Richard off already? Well maybe they think I've done it once, it's working for me so far, <laugh>, let's carry on. But it does seem as if people do carry on buying lottery tickets even if after they win. So you've got these 20,000 people who are potential in the pool to be able to win for a second time. Now, what are the odds of winning the lottery? Well, I gave a whole lecture on this last year, so I won't go into a lot of detail, but the most common kind of lottery is the pick six numbers from 49. That's the most common one still. Um, the odds of that are one in about 14 million. So I'm gonna say that that's my average summer, much more likely to win like the Polish lottery, it's one in 850,000 only to win. Um, you get a smaller prize, of course in the uk ours is more like one in 45 million, but let's take one in 14 million as a, as a reasonable average. So each of these 20,000 ja jackpot winners, they're doing their local lottery every week. So they have a hundred, uh, draws every year. Um, and and every time they do it, there's a chance that they might become a double lottery winner. Probably they won't. But let's work out the probabilities. Uh, so again, we'll do, like we did for the d and the one in a million shot, the chance of any one of these, uh, people not winning is one minus one in 14 million, right? So the chance that the probability that all of them will not win is that number to the power 20,000 multiplied by, by itself 20,000 times. And that's still very likely to be the outcome, right? Point nine nine eight six probability of nobody becoming the, uh, double jackpot winner that, that draw. But one minus that is the chance that someone will become, uh, a double lottery winner. So one minus that, that's about one in 700. So now all we have to do is okay, n equals 700, do my n 0.694 n trick rule of thumb. And we find that actually after 493 draws, it becomes more likely than not that someone will become, uh, a double lottery winner. So every few years we will expect to see this happen somewhere in the world. We obviously can't predict that it's gonna be evil and Marie Adams and the New Jersey lottery, but we can expect every few years somewhere in the world someone will have this incredible luck of winning the lottery for a second time. So it, it could happen <laugh>, it just pretty sure it won't happen to any of us, but nevermind, keep our hopes up. Now there's something, other, other thing that can happen in the lottery and it has happened at least once. And this is another sort of incredible coincidence. I'm gonna show you just a screen grab from the BBC uh, news website of the time of the event. The Bulgarian authorities have ordered an investigation. It says after the same six numbers were drawn in two consecutive rounds of the national lottery. So imagine the shock and panic <laugh> in the, in the Bulgarian lottery headquarters when, you know, their, their machine throws up exactly the same numbers as last time. And of course everyone will say, oh, the computer must be broken, there's something's going on, it's fraud. You know, this, this can't happen. This is, you know, there's no way this could have happened by chance. Um, I expect, you know what I'm going to conclude, but <laugh>, but let's, let's see, was there some awful fraud perpetrated for obscure reasons on the people of Bulgaria? Well, the chance of getting the same numbers as last week, it's the same chance as, you know, choosing the jackpot numbers correctly. One in 14 million right? Now that of course is extremely unlikely. If you look at, you know, all the lotteries going on in the world, it's still actually pretty unlikely, even if you do that, and you can, I did a quick calculation, I estimate it's about a one in seven chance of that happening of consecutive lottery draws coming up with exactly the same numbers, um, in the, in the 21st century. Well, which actually already happened this century in the next century. Uh, so, okay, that's unlikely. It's not impossible, but it's unlikely. However, if the same numbers had come up as like the week before or the month before, it would still have been a huge story and a big coincidence and kind of people would say there's something weird going on. So perhaps we are looking slightly too specifically at the very precise event that was the same numbers as the immediately proceeding draw. If we wind now net a bit, things start to look different. So by that point, uh, in, in time the Bulgarian lottery had been running for 50 years. So that's a hundred draws a year, 5,000 draws. Now if you start to think about, uh, in, in any draw, what's the chance of in all those 5,000 draws, two draws out of all of those having the same set of numbers? Well, the first thing you have to work out is how many pairs of draws there are. So if you're trying to find a pair, you know, draw A and draw B, well, there's 5,000 possibilities for draw A and then 4,999 for draw B. So that gives is our pair A and B. But if we just did that, we double count because pair a b is the same pair as pair BA, right? So actually we, we've double counted. So we have to halve that number. If we do, we still get a really big number, 12 and a half million pairs of draws that could potentially have the same numbers. And when you look, you know, the chance for each one is one in 14 million. But if you're having 12 and a half million opportunities for it to occur, it starts to feel much more probable. And in fact, the, the statistician, David Hand did a calculation around this. He said after 43 years, uh, you'd expect this to be, it's more likely than not that this would occur. So we were within 50 years of the Bulgarian lottery we're well into the time period where actually this is gonna happen at some point. So of course it was particularly noticeable because, um, it was consecutive draws, but it's, it's not massively surprising that it happened at some stage. And in fact, a similar thing happened in the Israeli lottery a few years later, but it was, I think they were three months apart, but everyone was still kind of a bit bamboozled by this happening. Okay? So we've seen that lots of different combinations, those different ways of combining lottery draws and looking at all the ways, uh, you could pair them led to the chances of the, the event happening being much higher than we might otherwise have thought. And it's the, the combinations and the number of possible combinations that gives a sound next way into amazing seeming coincidences that may be less amazing. And that's when we get networks involved. So the network I'm to talk about is, is the social network of human beings and how we, uh, know each other and are related and acquainted because we've all had the experience of you're being, you're in a, a social situation, you're at a party or something, you, you meet someone new and within a few minutes you've discovered, oh, what a small world. You know, my sister went to college with your cousin or something. Small world. We constantly say this to people as we desperately try and seek for links between us <laugh>, well, there are going to be links of course, because we all have friends and acquaintances and relationships, but it seems that these things crop up a lot more often than we might in our intuition might lead us to think. So let's, let's, let's think about this a bit. Let's take the United Kingdom as an example. Population, I don't know, let's say 70 million. How many connections do you have? How many people does each of us know? So if we think of like, okay, we've got friends, we've got family relations, we might have work colleagues, um, if you're like me, if you have children, you know your children's friends and you know your children's friends' parents, uh, you know, the, you know, the, the barista in your local cafe, you know, the local, uh, neighbors and I, I looked in my phone, I've got 414 phone numbers, okay? It's probably not, I don't know. Is that less or more than average? I've got loads of email contacts. So lots and lots and lots. If you were to try and write it all down, you'd get bored before you finished. I'm gonna estimate a thousand people, right? That I know, as in I know who they are. And if I were to go to a remote village somewhere and see them, I'd go, I'd have a conversation, right? That's my criteria. Oh, I know you, you, you, you live down the road from me. So let's say a thousand people that we know in somewhere, acquaintances. So that means if you were to pick a random person from the United Kingdom, the chances that that you would know them would be, well, I know a thousand people divided by the total population of 70 million, so one in 70,000, right? That's the chance that you know a particular person. But what what we actually do when we talk to, to strangers is we find mutual acquaintances or links between us. So what are the chances of having a mutual acquaintance? Well, for that, um, if you think of like, you know, here I am, I've got a thousand acquaintances and I want to know what the chances are that, you know, one of my thousand acquaintances, right? So we can do exactly what we've been doing for the past minutes and say, well, the chance of that happening is one minus the probability that you don't know any of these people. And if you work that out, you find actually the chance that we do have a mutual acquaintance is quite respectable, 1.4%. But the real magic happens if we allow an extra link in the chain, and we all know we are very keen to find links between it. We, we are willing to do this. So what we want to know then is the chance that a friend of mine knows a friend of yours, if you do that, it's almost certain to happen greater than 99% probability that one of my friends knows one of your friends. Now, the difficulty then is trying to find that connection in the five minutes you have to talk, but it's very, very likely that there will be a link, a path through this network that's really quite short. So that's when we say it's a small world. But I have highlighted some words on this slide that are very important, a little sneaky assumption I made, uh, which is totally invalid as an assumption. And that is if randomly distributed, just assuming that the chance of us knowing people is totally like random when we know it isn't. You know, there, if we live in the same town, we're more likely to know each other. Probably similar socioeconomic backgrounds, more likely to know each other. If you work in the same industry, you're more likely to know each other. So we know that human societies are very much not randomly distributed, um, in terms of who knows who. So that, that's not a good assumption to make. But all is not lost. So if, if we take a genuinely random network, right? So network has got nodes, so you can think of it, network or a graph, um, like the map of the London underground, it's got nodes, which in our case every node is person and then it's got edges. And those edges for us represent knowing someone in some way being acquainted, if that was completely random in any random network. And people's got end nodes, so end people are 70 million or whatever. And then how many edges on average are coming out of each node? So that's how many acquaintances we have. I took cake was a thousand. Then, um, the mathematics tells us that the average path length, which is the number of kind of the length of a chain from that links two people together, um, is given by this formula. Log in over long log. K, uh, doesn't matter if you dunno what that means. There's a formula that tells you what the answer is. But we do not live in a random network. Social, uh, human social interactions are not, not random. So is everything lost? Do we abandon this? Fortunately not. Um, there's a brilliant bit of, uh, research and a paper in nature by, uh, Duncan Watson, Steve STRs, um, what's now called the Watts STRs model, which looks specifically at the kind of networks, um, that include human, you know, networks of society. Um, they call them small world networks for the obvious reason. Now, they are definitely not random. They have lots of clustering and very much not random, but they found that as long as you have just a few, uh, circuit breakers they call them. So people in our, in our example, people who have lots and lots of links who know people from, you know, all walks of life, social hubs, um, there's one in my circle, she's called Alex. She knows everybody from every single walk, walk of life. And because I'm her friend, then I'm linked to all of those people. So even if I'm a bit shy, you know, I, I have links, I can still be on this, this graph, um, just a few of those circuit breaker people can cause this network, even though it's really not random to behave like and have similar properties to, to genuinely random network. And so to a really good level of approximation, we get this same average path length even in this non-random situation. So that's really cool. And if you put in the numbers that we've got, so in my UK example, if you put in the 70 million and knowing 1000 people, you find that the average path length, so the number of connections you need is just 2.6. So it kind of fits in quite well with our rough calculation we did before. And even if you take the whole world, so 8 billion people, let's say population, and let's have a really conservative estimate for the average number of, of links that there are average number of acquaintances.'cause you know, some people live in remote mountain villages, some people live in big cities. Uh, newborn babies are really letting the side down in terms of how many friends they have. So let's take an average of 100 that ought to, you know, take account of, of those variations. Even if you do that, you still get an average path length of 4.95, less than five links in the chain. So the six degrees of separation theory that we've all heard of actually really does hold water mathematically speaking. And it's because of all these ways that connections can happen, the power of all these combinations allows these seeming coincidences to happen much more often than we might intuitively think. Okay? So I've spent half an hour saying it's not well coincidence. It's, you know, it's all coincidence. It's gonna happen anyway. Is am I ever going to say something's more than coincidence or am I just gonna <laugh> explain why we should expect these coincidence exciting as they are, but they don't necessarily mean anything? Well, sometimes things are more than coincidence and that's when we can start to think scientifically. I'll give you just one example, very famous example from history, uh, John Snow, uh, investigating cholera. And this is a little map, very famous map, which shows that there is a cluster of cholera cases in a particular area near a street called Broad Street, not too far from here. And so he notices this, maybe it's just a coincidence. Is it a coincidence he didn't think it was what could be causing it? That's that's when we started to think scientifically. We say this is a strange pattern. Maybe it's more than coincidence. But then you have to think of or try and think of a reason why this might happen. What is causing it. It's not just, oh, divine intervention has decided, you know, eventual God does not like the inhabitants of no, there must be a reason for this. What's the reason? And he, he realized, okay, there's there's a water pump just there on that street. And so then he has this theory that maybe cholera is waterborne and not airborne as previously thought. So he's got now got hypothesis to explain this seeming coincidence, but it's not a coincidence. Then you have to test your hypothesis. You can't just stop there. How do you test the hypothesis? You remove the handle of the water pump, then no one can access that water from that. Well, and guess what? The cholera cases went away. Went right down. And so he's got, he's noticed something. He thinks it's not a coincidence. He forms a hypothesis, then he tests the hypothesis that's doing science. There are lots of people not doing science in the world. I have to tell you, um, right in astronomy for hundreds of years, of course thousands of years astronomers have spotted patterns, the ways in which planetary bodies behave and tried to understand and explain them formed hypotheses, tested the hypotheses. That is very different from saying, Hey, the moon exactly covers the sun during a total eclipse. That's really spooky. And we know the moon is smaller than the sun, but the moon is 400 times smaller than the sun, but the sun is 400 times further away than the moon. Whoa <laugh> that. So that yes, it's a really cool thing, but without some sort of reason as to why that might happen, why that would have to happen in some way. We are just saying it's a nice coincidence and there are very, very many astronomical bodies and there are many measurements that we can take about them. And when you have lots of numbers and you can play with them and manipulate them in various ways, it's quite easy to find things that appear to link to each other. And so there's some of this I've, I've investigated, so you don't have to, but there's online, there's a certain amount of stuff that isn't really very, uh, correct, <laugh>. Sorry if I'm breaking this to you for the first time. I just wanna give you a couple of examples. And this is the kind of thing that can, can fool people into, into thinking there's something deeper going on. But really it's just the fact that there are so many ways to combine numbers. So here is what's supposed to be an amazing sort of, you know, sacred mystical alignment of something or other, right? What do you do? You take the length of the earth year in days, multiply by the length of the lunar month. Why are we multiplying? Don't know. Then you multiply by the golden ratio, the lovely phi, wonderful sort of mystical sacred thing. Why are we doing that? Don't know. Then you divide by pie, another lovely mathematical consonant. Why are we doing that? Don't know. And the answer is that you get, it's not a whole number, but to the nearest whole number. It's 5, 5, 5, 5. Are you impressed? No. Weird. Okay, but no wait, I haven't got to the best bit. Let's bring a pyramid into this.'cause they always have to come in, right? Pyramids, like here's a cross section, great pyramid of Giza. So it's got a triangular cross section there. We're not gonna take that triangle, that would be too easy. We're gonna take the equilateral triangle that just fits inside inscribe, an equilateral triangle. What is the side length of that equilateral triangle? 5 5, 5 0.5 feet. Amazing. So, and obviously you are stunned and amazed by this, uh, mystical phenomenon. There are some questions we might have about this, um, which we'll we might think about if that isn't impressive enough yet. I'll just give you one more amazing thing, okay? Draw a circle now inside the pyramid and the inscribed circle has the diameter just the same as the length of the year. It's amazing. So there are some questions, it's questions. One most important question. Why are we measuring this in feet?<laugh>, ancient Egyptians did not use feet. Aliens probably do not use feet. Feet as the name suggests, uh, derived from the foot, which is a human thing.<laugh> not used by the ancient Egyptians or probably aliens or anyone else. Um, so why, why are we suddenly deciding we're gonna measure things in feet? Uh, if you want to draw a circle that represents the earth in some way, why don't you make its diameter the damage of the earth and not some other thing. And then that little asterisk, that fair to say, um, in fact the diameter isn't that it's two feet more. Uh, but as the writer of the website I got this from says, um, that probably represents the Earth's atmosphere, those extra two feet <laugh>, it's fine, it's gonna be fine <laugh>. But of course then it immediately negates the spurious precision of these, you know, four decimal places. But oh, but two feet plus or minus in either direction, you know, completely negates this. So it's really easy. As I've said, there are a lots of numbers in the world, mathematician, telling you there are lots of numbers and you can manipulate them in lots of different ways and combine them. So really, of course you can make these things. And just to emphasize this, I want to present to you my very important findings on the Mystic Gresham numbers.<laugh> Thomas Gresham, founder of Gresham College. I think he may have been able to predict the future. So look at the right 1, 3, 5, 7, and nine. The important very important sacred odd numbers have lots of different meanings in different civilizations and traditions. I'm gonna show you a triad of ways that these five numbers, uh, have, are important to the Gresham legacy. I should put a bigger thing on this. This is all, all nonsense. I'm telling you right now, be aware of it all the way through, right? So 1, 3, 5, 7, 9, the first three that we're gonna see, you can already see. And there are three stars on the coat of arms there. So that leaves 1, 5, 7 and nine. Oh my goodness, that's the year that Thomas Gresham died. Amazing, right? We want two more sets of 1, 3, 5, 7, 9. If you take that 1, 5, 7, 9, just flip the two numbers, the last two numbers over 1, 5, 9, 7 50 97 the year Gresham college was found it amazing, right? If you take still the 1, 5, 7, 9, now flip the five and the nine over. This is where it gets really spooky. Or you ready? Let's see, I was born<laugh>, I I'm your laughs are laughs of all I know. So, but hang on there, there, there's, I promised you three sets of these numbers. So where are the other two threes? Well, I'm the 33rd Gresham press of geometry. So, okay, now you are impressed, right?<laugh>, so did Thomas Gresham. No, surely maybe this is an indication that he knew he had a plan that was all accommodating in this lecture right now, which is happening on the fifth day

of the third month at 1:00 PM And if you don't want me to speak for 79 minutes, we better leave it there. But I wanna, I wanna, okay, this is all just to say, you know, some of the, this is clearly silly, but sometimes, you know, all of this stuff with the pyramids and, and the cosmic coincidences, you know, it's one can get drawn into these things, but you can really easily, I, you know, just in a few minutes you can really easily do something with numbers that on the surface looks sort of good, but it doesn't really hold water, doesn't make sense. Um, and okay, finding all these things about the pyramids is silly but harmless unless, you know, we think of the effect on the blood pressure of real archeologists, but, but there are some coincidences or the misunderstanding of coincidence can have really serious real life implications and can change people's lives, um, and affect them very, very seriously. And I wanna just talk for a few minutes more seriously about the use of statistics, probability, misunderstandings of coincidence in legal cases. I wanna give you a general example first, uh, which is a prevalent thing and it's even got a nickname, the prosecutor's fallacy. And this is, well, I'll give, just make up an example. Um, let's say there's a crime committed and you get some sort of forensic evidence, some partial DNA or blood marker or something that you find at the scene of the crime, and you then run that through your database of known criminals and you find a match, you go, okay, well that's clearly the guy. So there's, you know, court case and when it comes up and the prosecutor will say, well look, we found this match and they might even put some numbers, and we all know numbers to tell you lots of things. They might say, only one in 10,000 people have that particular marker in their DNA. So this guy, look at him looking shifty already, he must be him <laugh> only one in 10,000 people. What are the chances? Um, it must be this guy, but they are, that's the wrong number to think about. What we really want to know is given that someone has that DNA marker, what is the chance then of them being guilty or, or innocent? So let's say it is one in 10,000 people in London, there are 9 million people. So that means that in London we'd expect 900 people to have this particular DNA marker. Okay? Well that means if we pick a random person from that population of 900 people that have this thing in the, in their DNA, then the chances that we want to know what are the chances that they're innocent given that they have that marker, right? That's, that's the question we want to know. And of course, there's somebody in those 900 is guilty, um, but the other 899 are innocent. So if you pick a random person with that, with that particular DNA or blood marker, then there's a 99.9% chance they're innocent. We'd have to have concrete evidence of another kind that the person, the suspect actually had been involved in some way. There's some reason to think they might be involved in the crime before this, you know, before we can make any conclusion from that. So high likelihood of them still being innocent, okay? I'm gonna give you a real example now, and this is just a shocking miscarriage of justice that happened a few years ago. You might remember, uh, the case of Sally Clark. So Sally Clark, um, it's really a terrible thing that happened. So she had a baby and very unfortunately, um, the baby died suddenly. So it's called sudden infant death syndrome or sometimes called caught death. So the baby died, it was tragic, and then a couple years later she had another child. And very sadly, that baby died as well suddenly, and people got suspicious and she was accused and tried for murder. They said she must have smothered the babies. This couldn't happen by chance. And one of the key, uh, witnesses for the prosecution was a pediatrician called Roy Meadow. And he said these three things in his testimony, he said, okay, the chance of a SID'S death in a family like that. So affluent, everyone's healthy, nobody smokes, is about one in eight and a 5,000. Okay? Then he said, that means the chance of two deaths happening of Sids in a family like that is one in eight and a half thousand squared, which is one in 73 million. And that would only occur by chance once in a century. So there's, there's a lot wrong with that. There was a immediate outcry after this, perhaps not immediate, but there was an outcry and people, um, there was an appeal against this. The biggest problem with with that is the middle statement, right? That the odds one in eight and a half thousand is probably about right, but he made the assumption that these things are independent of each other, and that unfortunately is not the case. We don't quite know why. But in a family where there has already been one Sid's death, actually it's about 10 times more likely than for the general population that there will be a second. There may be some genetic factor, we don't quite understand what it is, but the chances of raise, so actually it's not one in 73 million, it's more like one in 7 million. Now you might say, okay, well that's still really unlikely. So, you know, she probably did still do this, but again, we are not comparing the right things. If you compare the probability of two SID'S deaths, um, one in 7 million with the probability of the babies not dying, then yes, but what we have to compare is the probability of two Sid's deaths with the probability that she murdered her babies, right? And that's much, much lower. Um, so yeah, seconds is death 10 times as likely. But, uh, if you look at the, and they did do this, if you look at the data for the probability of, um, two babies dying of suddenly infant death syndrome compared to the probability of two babies being murdered by their mother, of course the, well, I hope it's of course the Sid's deaths much more common than than than the other possibility about nine times more common. And so these statistics sort of glibly used and, and got this woman convicted of murder to add to her other just awful ordeal that she'd experienced. And this just shows, you know, we, it's so important for people to understand a bit about probability, a bit about statistics, a bit about when things are coincidences and how often coincidences can happen in order that we, you know, we can have a, a conversation, a kind of numerate conversation in these court cases and understand what the implications are and what's really going on. So that was okay. That was pretty somber for me, I thought. Um, you know, it's important to say, but I would like to finish on a slightly lighter note. Slightly lighter note to send you off into your days, um, um, you know, with something a bit more light on your, on your minds. So let's think about an amazing coincidence that happened. A very astonishing thing. So a few years ago, and there was an episode of the BBC's program. Who do you think he was? If you're not familiar with this, uh, it's a kind of family history program. They get a celebrity and they look at their family history and they say, oh, you know, you've related to, oh look, 200 years ago, so and so was a scullery maid and someone else served on the whatever ship. And they give them an interesting bit of information about their family history. So in 2016, there was an episode with, uh, the actor Danny Dyer, and there was an amazing revelation they showed in this big scroll. He can look at his look, go back so and so, so, so, and then royalty. He was directly descended from royalty. Remember, I think it was Edward iii. And then if you go back, William the Conqueror, right? So Danny Dyer directly descended from William the Conqueror. Wow, how exciting is that? I thought, well, what are the ch you know, maybe some of us might be related to royalty. So let's think about this. Our family tree, and we all know what a family tree looks like. It's us, two parents, four grandparents, eight great grandparents. I can't stretch any higher than that, but it doubles and doubles and doubles each time. So if you go back 30 generations, about 900 years by that point, we've got in that family tree, we've got two to power, 30 empty boxes that we have to fill. So how many names is that? Well, it's over a billion. It's over a billion right now. There's an immediate, practical issue there, and that is the population of the entire world in 1100 ad is only 320 million estimates say. So how on earth can this be, right? Well, of course what has to happen is that the same people are gonna have to perform more than one role in the family tree. So, you know, your great-great-great-great, great great aunt, might have been the same person as your great-Great-great-great great grandmother. Um, and so if we go back far enough, this kind of nice bifurcating thing actually starts to become a clustered, interlinked tangle of multiple, uh, relationships and, and connections on your family tree. And that's not just your family tree. Actually everybody, if you go back, those 900 years has got those billion spaces in their family tree, which are gonna have multiple people in multiple places. And the same 320 million people, or even fewer than that, if we look at European ancestry, uh, are gonna have to be on lots of, lots of family trees. And the extent to which this is the case is really quite startling. So a mathematician called Joseph Chang, um, wrote about this and he describes something called the identical ancestor point. So in Europe, the identical ancestor point is between 800 and a thousand years ago, let's say 900. That's about between those numbers. So what is this point? Well, at that point, so you can show that about 20% of the people alive in Europe back then actually haven't got any living descendants. Either they didn't have children or their children didn't, their family line died out. But for the anyone in the remaining 80%, let's say William the conqueror, we know he's got at least one living descendant, Danny Dyer <laugh>. So he, he is one of the people who has at least one living descendant. It's amazing, but true. It's really counterintuitive. But of those people alive back then, if they have any living descendant at all, then they are the direct descendant of everybody alive now with European ancestry, everybody. So we are all descended from, from royalty, right? So we're all descended. If we have European ancestry, we are all descended from William the Conqueror. And actually you can take this worldwide, you have to go back a bit further to cover everyone in the world. But the world identical ancestor point is about 3,600 years ago. So anyone alive in, uh, 1600 BC who has any living descendants is the direct descendant of every single one of us. And that's really amazing. So on that happy note, your majesties, I shall finish for today. Thank you. The problem with the six degrees of freedom is simply that you don't know who to pick. Yes. If you want a connection with Richard Nixon, right, who would you pick? Well, you've got no idea, but you've probably only got two degrees of freedom away from Yeah, yeah, yeah. Anyway, and, and so I wondered if you would talk about that aspect of statistics. Yes, you are right. It's knowing how to find these links. And of course there are, there are now there are plenty of ways we can do this online. I mean, there, you know, maps of the internet, and that's how the, you know, Google and other search engines work by knowing what the links are. They've got, um, it's based entirely on, on graphs. These networks, uh, the, the search engine will look through and look at what's linked to what, which pages are linked to others. So if you have a big, you know, database, you can search these things and try and find these shortest parts. Of course, we don't have that in front of us, but there was an experiment quite a while ago, this, you know, before the internet, but maybe in the sixties, there was an experiment where people were given a name of someone. This is in America, they were given just someone's name, uh, and you know, so and so, South Carolina, whatever. And they had to get to them by, send a letter to them by sending it to someone they knew who might know this person. So kind of try to exploit who might be likely to be able to get this to this person. I never know. And they found that actually, and of course this is within imperfect information, you're just making educated guesses. But they could do that usually within four steps because people were actively trying to think about how it might work. So if we can try and engineer these things, as you say, and then we can improve our, our chances of being aware of the links, but the links are there, I'm fascinated by origins, uh, and proba probability in terms of science, uh, particle physics, et cetera. Mm-Hmm.<affirmative>, uh, the fact that the universe began and the fact that life began and it ha has happened is a probability of that one. Mm.<laugh>. I mean, yeah. So, so this is, we could have talked a bit about the philanthropic principle, right? That says it's amazingly unlikely on the face of it that, you know, there's this planet that's just the right distance from a sun of just the right temperature that's not, you know, unstable or stable. And then we've got the moon that helps us not to be hit by asteroids constantly. And there are tides and there's water and the temperatures right in this goldilock zone. So on the face of it, gosh, that's awfully unlikely. But on the other hand, it had to have happened, as you say, probability one'cause we are there to ask the question, right? So it's kind of, given that it's happened, what's the probability that it's happened? One <laugh>, you know? Yeah. It's, you, you probably need to find Chris Linox next book, which is called The Accidental Universe. Perfect.<laugh>, talking About exactly that point. It's all planned.<laugh>. Next question over here. Thank You. Um, hello. So as a sort of random aside, I remember getting on a tube journey once and in three carriages, one after the other. So one line after the other. I, I knew someone on each carriage, which always struck me as it's quite improbable. But I'd like to know, when you see a random statistic, something that seems absolutely extraordinary, what's the thought process to go through in a sort of lay sense? You know?'cause it seems there's like errors of assumption. Yeah. That is just the sense of the scale of something, you know. So what actually do you think about when you see these things? So I might, I mean, the sort of general term might be what's the denominator of the <laugh>? You know, so how many times have they of this test been done? So if you see an advertisement that says, you know, um, all these people tried this amazing beauty product and 18 out of 19 said it was great, but how many sets of 19 people have they asked <laugh> in order to get that? So yeah, what's, what's the pool of, of possibility for this outcome to have happened? And that, that's part of it. Um, and, and of course the more, the more trials you have, the more likely it is to occur. Also, I think thinking about the, the near miss possibility. So if are, if someone said, well, this is close enough, so that's another way to get things happen, much more likely. So if you make some prediction and you were nearly right, <laugh>, uh, I, I'm gonna try and predict a random number and I'm gonna say it's 54 and you know, oh, well it was 50. Okay, well, near enough, then you massively increase your chance of getting it right. So you, you can be interrogating the, the reported output. You know, are they, how many trials were there before this successful one? Uh, how close is it to what you really were thinking of? Those are the, those are two really key things I think to remember. Um, and just to be aware, you know, there's, with the more combinations you allow, the more likely, as we saw with my very exciting revelations about Thomas Quish <laugh> Ladies.

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