problem, but what is the probability that we will experience more than one once-in-a-lifetime (Poisson-distributed) event in our lifetimes?

David Hand (DH): The improbability principle answer is that you’re certain to experience more than one once-in-a-lifetime event in your lifetime. The point is that there are all sorts of once-in-a-lifetime events going on all the time. Find enough of them and it becomes as probable as you want that you’ll experience more than one, and that you’ll experience one type of such event more than once.

Alternatively, a more conventional answer is that, if I define a particular kind of once-in-a-lifetime event as an event which arrives according to a Poisson process with mean inter-arrival time equal to the average lifespan, then the probability of experiencing more than one such event during an average lifetime is about 0.26. You don’t need many such series of events to make it overwhelmingly likely that one of those series will have multiple events in a lifespan.

“Things which ought to be expected can seem quite extraordinary if you’ve got the wrong model”

In July, Phillip Watkins, a biostatistician at the Texas Tech University Health Sciences Center, wrote to Significance, posing something of a riddle: “What is the probability of experiencing more than one once-in-a-lifetime event in your lifetime?”

The question had us intrigued – and stumped. But we figured David Hand might know the answer. Hand is senior research investigator and emeritus professor of mathematics at Imperial College London, and the author of the book, The Improbability Principle, which seeks to explain why extremely unlikely events are actually fairly commonplace.

Over a series of emails, Hand and Watkins delved into the detail of the book, the principle itself, its law and implications. But first, the answer to that riddle…

Phillip Watkins (PW): David, I wanted to start by asking whether your improbability principle has something to say about the question that led to this interview. Admittedly, it’s more of a probability

Phillip Watkins talks to author and statistician David Hand about his new book, The Improbability Principle, and why the chances of experiencing two once-in-a-lifetime events are better than you might think.

© 2014 The Royal Statistical Society36 october2014

There are probably other ways of defining a “once-in-a-lifetime” event, but the general principle will hold for all such definitions.

PW: What is the kernel of your improbability principle? That is, why do these supposedly unlikely events actually happen more frequently than we tend to expect them?

DH: There are five fundamental laws which intertwine to make extraordinarily improbable events commonplace, which is a statement of the improbability principle. These laws then interact with the human mind, with how we think about things. The laws are: the law of inevitability; the law of truly large numbers; the law of selection; the law of the probability lever; and the law of near enough (see box on next page for details).

PW: What inspired you to sit down and write this book?

DH: There wasn’t one particular event which inspired me – just the amazement we all feel when we hear of incredible coincidences or unlikely events happening. The sense that surely the odds must be impossibly against someone winning the lottery twice, or accidentally finding herself seated next to someone with the same name. There’s this nagging feeling that perhaps there’s something about the universe that we haven’t quite

grasped; that perhaps something or someone is manipulating events, steering them into these startling collisions. Coincidences are fun, they make you sit up and take notice and say, “There must be more to it”. And that’s what inspired me: the wish to dig deeper and explore this apparently counterintuitive aspect of probability.

PW: I’m glad you did, as it has helped me better understand coincidences – including one that I call the “new car syndrome”: the phenomenon of purchasing a new make/model of car and then seeing that same car everywhere.

DH: This is rather like the experience of encountering a new word, and then coming across it again in quick succession. Of course, the “new car syndrome” is compounded by manufacturers marketing particular models at particular times. Whatever induced you to buy that make/model is likely to have had a similar effect on others. And then you naturally have a tendency to take note of that type of car. You might well have seen more cars of a different, but equally rare type, but failed to notice them. After all, they had no special significance for you, so why should you? This is the part of the improbability principle arising from the way the human brain sees and processes the world: our attention is naturally drawn to things which have relevance for us.

PW: I agree with you that coincidences can be fun, but some have grave consequences, like a recent plane crash in Georgia that hit and killed two pedestrians walking on the beach. What does the improbability principle say about such a seemingly unlucky chance occurrence?

DH: Fortunately, plane crashes are rare – around 150 per year, which is tiny compared to the number of flights – of which there were around 32 million in 2013, according to the International Civil Aviation Organization. But even 150 per year is enough for us to expect that occasionally they will crash in towns or places where there are people. And, of course, when they do, they tend to hit the headlines. Your question immediately reminded me of the helicopter which crashed onto a packed bar in Glasgow, Scotland, in November last year, killing ten people, and

the helicopter which clipped a crane on a high-rise building in London in January last year, killing both the pilot and a passer-by on the ground. Unfortunately, the improbability principle says we should expect this sort of thing to happen, and it does.

PW: If we should expect small probabilities to happen, does this cast any doubt on the use of small p-values to assess significant relationships? If so, what are the alternatives?

DH: I think the use of small p-values is very much in accordance with the improbability principle. Understanding of the principle tells us that if we see an incredibly rare event occur then we should explore which of the laws constituting the principle apply. To take an example, the law of the probability lever tells us that even slight changes in our beliefs about the world can have dramatic effects on probabilities. We can then choose between competing explanations by balancing the probabilities of events if each of those explanations were true.

PW: I understand the above method to be a variation on proof by contradiction, using an unlikely chance occurrence as proof against the null hypothesis. In fact, I’ve heard the frequentist viewpoint more generally summarised as “the truth is out there (and I can know it)”. As such, do you think the Bayesian methodology is better suited to the scientific method? If so, why do you think scientists are so hesitant to adopt it?

DH: That’s a nice characterisation of the frequentist approach to hypothesis testing – though that school of statistics does more than merely test hypotheses. Incidentally, I hesitate over the word “proof ”. Statistics doesn’t claim to “prove” anything, but rather seeks to show how evidence sheds light on different propositions.

In general, I try to avoid summarising any particular school of inference in a soundbite since – although such sayings have a grain of truth – their inevitable simplification means they’re bound to mislead (“Bayesians say you can believe whatever you want, as long as you’re coherent”). My perspective is that the right approach to a problem depends on that problem, on what you want to know and what question you want to answer. I’m afraid all schools of

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interview

inference, including the multiple Bayesian and frequentist schools, have issues at their heart which mean too zealous a generalisation about their applicability should be avoided.

PW: Returning to the aforementioned law of the probability lever, I am reminded of the importance of checking the assumptions of any statistical model. I feel people often overlook the dependence of events when they utter the incredulous phrase, “Wow, what are

the odds?” How large a role does the false perception of independence play in your improbability principle?

DH: In technical statistical terms, that’s exactly what this law shows – things which ought to be expected can seem quite extraordinary if you’ve got the wrong model. I give many examples in my book, but familiar ones are extreme financial events. If you base your thinking on a normal distribution model,

then incredibly unlikely things seem to occur. But choose a fat-tailed distribution and they become all too frequent.

It’s difficult to divide up the contribution of the five constituent laws of the improbability principle, but it’s certainly true that the unjustified assumption of independence is one way in which the law of the improbability lever manifests itself. Charles Perrow’s theory of normal accidents is based on the notion that insignificant

David Hand on the five laws of the improbability principle

1. The law of inevitability – something must happenThe law of inevitability says that one of the complete set of all possible outcomes of a random event must occur. So, to see this law in action, we need to be able to list all the possible outcomes, at least in principle: the set of all possible lottery tickets that might come up, the set of all birthdays in a year, and so on.

2. The law of truly large numbers – with a large enough number of opportunities, any outrageous thing is likely to happenIn July 1975, a taxi in Hamilton, Bermuda, knocked Erskine Lawrence Ebbin from his moped, killing him. The year before, his brother Neville Ebbin had been killed by the same driver driving the same taxi and carrying the same passenger while riding the same moped on the same street. The Global Health Observatory puts the number of annual road traffic fatalities at about 1.24 million. With that number to choose from over time, it would be surprising if we did not see coincidences of the kind in the Ebbin story.

3. The law of selection – you can make things as likely as you want if you choose after the eventUS presidents Abraham Lincoln and John F.

Kennedy were both assassinated on a Friday, and both in the presence of their wives. Both were shot in the head from behind. Each had a son who died while they were president. Lincoln had a personal secretary named John, and Kennedy had one named Lincoln. Lincoln became president in 1861 and Kennedy in 1961. Both were succeeded by presidents named Johnson who – wait for it – were born in 1808 and 1908, respectively. They each had four children.

The law of selection is manifest by the selective nature of the matches listed above. These were selected from a large number of potential pairs, most of which did not match: for example, the names of their mothers, the birth dates of their mothers, the heights of their wives, and so on.

4. The law of the probability lever – slight changes can make highly improbable events almost certainIn 2005, one of my graduate students and I flew to the US to attend a conference. We had booked our seats separately, but we found ourselves seated next to each other. 747s normally seat somewhere between 400 and 500 passengers, so the probability of us being placed next to each other by chance looked like about 1 in 450. But we were travelling

in the same class, so the number of available seats was less than 450. Also, few seats occur in isolation, so you have to sit next to someone. Furthermore, many – perhaps most – people do not fly alone, so the number of single seats is far fewer than appears. Put all these things together and they dramatically change the probabilities.

5. The law of near enough – events which are sufficiently similar are regarded as identicalIn the 1950s, Eric W. Smith, of Sheffield, England, was in the habit of collecting horse manure for his tomato plants from the woods behind the house. One day he saw another man doing the same. When he sat down on a bench to rest, the other man did the same. His name? Also Eric W. Smith. Further discussion revealed that the first Eric W. Smith had the middle name Wales, while the second had the name Walter. So it was not an exact match – but near enough to be surprising. But what if they had different middle initials? Now it is not an exact match, but is it still surprising? How far can we relax the conditions for a match before we are surprised by it? How near does it have to be?

Extracted from The Improbability Principle website,

improbability-principle.com.

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errors or flaws can combine in the complex interacting systems and machines which characterise our society to lead to major disasters. And, in fact, the false assumption of independence was one of the factors which led to the 2008 financial crash. The aggregate consumer housing debt risk models used prior to the US subprime mortgage crisis assumed independence of the default risk of individual mortgages, when the mortgages actually had highly correlated creditworthiness, via the common factor of inflated property values.

PW: In my work as a statistician in clinical research, I see the laws of the improbability principle playing themselves out in several ways: many investigators will look at numerous potential explanatory factors while assuming a normal distribution/error, and those with significant findings will be more likely to publish. As such, how can statisticians combat these abuses in statistical application to help minimise the Type I error in published manuscripts?

DH: I agree – the laws do manifest themselves all over the place. Carry out enough significance tests and you should expect to see some significant results, even if there’s no underlying effect. Statisticians can help to avoid that particular problem by using multiple testing controls, like false discovery rate methods. Likewise, the tendency to publish significant results and leave the non-significant ones in the file drawer can be tackled by rigorous honesty. But, of course, this doesn’t completely solve the problem – something I discuss in the book. Statisticians know how important it is to develop hypotheses on one data set and test them on another, but sometimes it’s not clear whether a data set has already contributed to the idea for a hypothesis. In general, while statisticians are better placed than most people to spot when an event or phenomenon arises

as a consequence of statistical artefacts, such as the laws of the improbability principle, even we can be fooled.

PW: The subtitle of your book is ‘Why coincidences, miracles, and rare events happen every day’. What are the differences, if any, between those subcategories of improbable occurrences? Is the relative size of the associated probabilities the only difference, or is there something more to it?

DH: There are no real physical or objective differences. They’re all simply rare events. But the way we look at them means we may see them as different. A miracle, of course, occurs when we attribute a supernatural cause to such an event, instead of looking at the improbability principle and doing the maths. A coincidence is when two or more things appear to come together purely by accident – like Anthony Hopkins finding, on the seat next to him, a copy of the book he’d failed to find in his search of London’s bookshops. But they’re all improbable events.

PW: I see. So when you say “miracle”, you mean “a fortunate occurrence ascribed to a supernatural force”. Others might define a miracle differently – like Albert Einstein, who suggested that there are two ways to live: as though nothing is a miracle or as though everything is a miracle. After writing this book, with which approach to life do you more strongly identify?

DH: Interpreting Einstein’s use of the word “miracle” as “something wonderful”, I think I identify with the latter – live as though everything is a miracle. After reading the book, people sometimes ask me if understanding these amazing coincidences makes them seem pedestrian and ordinary, and if I’ve lost my sense of wonder. My answer is that the opposite is the case, and I

sometimes illustrate that by reference to the rainbow. The fact that we understand the physics by which light reflects and refracts around raindrops doesn’t make it any less awesome when we see a rainbow arcing across the sky. On the contrary: understanding the complexity and elegance of the mechanisms that lead to such things occurring makes them much more awe-inspiring and wonderful than merely saying, “Gosh, how amazing”.

PW: You write that the tendency to attribute rare or unexplainable events to a supernatural power is a perhaps a poor explanation, but is chance really better? That is, couldn’t we just shrug our shoulders at any sufficiently complicated system and say, “Beats me; it must be dumb luck”?

DH: I think chance is better. The point about chance is that while we might not be able to say what the outcome will be for any particular event, when we start to aggregate events we find the aggregates have patterns. While I can’t tell you whether the coin will come up heads or tails, I know that, if I toss it 200 times, the proportion of heads won’t be too far from 1/2. And exactly the same applies to other things. While I might not know if this medicine will be more effective than that for you in particular, I might well know that the first has had a higher success rate than the second on people in the past. In short, chance has laws which we’ve slowly understood, and we can use those to limit the range of things which are likely to happen in the future. You can’t say that sort of thing if you appeal to a supernatural explanation, where anything goes.

The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day, by David Hand, is out now, published by Scientific American/Farrar, Straus and Giroux. See improbability-principle.com for more information.

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