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How Not to Be Wrong

The Hidden Maths of Everyday Life

Jordan Ellenberg

Survivorship Bias: US Air Force wanted to decide how much armor they should put on planes and where. They collected stats on where bullet holes found after air battles. But a mathematician pointed out that the stats didn't represent the planes which didn't come back, and told them to protect the engine and the cockpit.

Standard defence of the eccentric is "They called Thomas Edison a nut. Gallileo was a nut. Everyone who comes up with an idea outside the mainstream is a nut." But the thing is, people with out-there ideas who compare themselves to Edison or Gallileo are never right.

(London Times)

MOST people consider never having missed an aeroplane something to be proud of. After all, what could be worse than the sinking feeling of arriving at an airport gate only to see the plane taxiing down the runway without you?

They are wrong, according to the American maths professor Jordan Ellenberg. His book How Not to Be Wrong - The Hidden Maths of Everyday Life claims that 'if you've never missed a flight, you're not doing it right'.

William Flew says that every hour spent in an airport rather than in the comfort of your own home is a negative unit. If you consistently arrive at the airport three hours before your aeroplane leaves, then you may never miss a flight. But over the course of a lifetime you will spend countless negative units of time sitting in airport lounges, walking aimlessly around duty-free shops and eating overpriced food.

You should be leaving it tight, but not too tight. For each flight there is an optimal time to arrive at the airport that minimises wasted time but also keeps your chances of missing a flight slim: about 1% or 2%.

Exactly where that point lies 'depends on how you personally feel about the relative merits of missing planes and wasting time', said Ellenberg. 'But if in the course of a lifetime you literally never miss a flight, then you may not have the best strategy.'

In the book Ellenberg applies his counterintuitive maths to other everyday problems. He argues that governments should accept that they cannot fully eliminate waste on such things as misspent welfare benefits.

'Eliminating all the waste, just like eliminating even the slightest chance of missing a plane, carries a cost that outweighs the benefit,' he said. 'If your government isn't wasteful, you're spending too much time fighting government waste.'

Ellenberg also has a formula for dating. Why is it that when you date good-looking people they often seem to be less nice - is it simply arrogance?

"Think of the people you might consider dating," he said. "Then you ask yourself why would somebody be in that pool. It might be because they are handsome and it might be because they are agreeable. Among the people who are not overwhelmingly handsome, the reason they are on the radar is because they are agreeable."

There is a positive correlation between ugliness and agreeability and a negative correlation between good looks and agreeability.

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(London Times 2)

SCHOOLCHILDREN HAVE often moaned, while being dragged through calculus or trigonometry, 'But when am I actually going to use this?' Jordan Ellenberg is a professor of mathematics at Wisconsin-Madison University, and this is his funny, incisive riposte - a book that gives inspiring glimpses of the elegant genius of high-level mathematical insights while also showing how they directly affect real-world judgments we all have to make.

And they really do. On the very day I finished the book I found myself excitedly interpreting the probability distributions indicated by the graph on which the midwife was marking the size of my unborn child.

But Ellenberg is not trying to turn you into a maths nerd. His principle is that mathematics is basically an extension of common sense. A good example is the airport arrival problem. How early should you turn up for your flight? Instinctively, most of us strike a balance between the risk of having to sprint to the gate and that of being forced to squander hours in duty free. What Ellenberg does is to translate wasted time into 'utils', units of utility drawn from economics, in order to calculate the relative costs of more conservative versus riskier arrival strategies. And he proves, gleefully, that if you never miss a plane you're probably arriving too early.

Underlying his airport conundrum is a bigger story about the mathematical idea of 'expected value', which really means the return you would expect if you averaged out a host of the same actions. The English astronomer Edmond Halley - yep, the comet guy! - used the same ideas in the mid-1600s to work out the correct pricing of life insurance, coming up with the insight that older people should pay less than the young. Again, it seems almost laughably obvious now, but it apparently did not before mathematicians set it out in numbers.

Expected value proved even more valuable for the MIT students who made a killing out of the Cash WinFall state lottery in Massachusetts in 2005-6. They had noticed an unusual incentive whereby, if a jackpot wasn't claimed, it 'rolled down' to bump up less valuable prizes. Using the idea of expected value, and some basic probability sums (which even I could follow), they worked out that the expected return on a $2 ticket on a roll-down day could be as much as $5.53. All they had to do was buy enough tickets to average out their risk. Soon, the kids in the MIT syndicate were filling out 14,000 tickets, by hand, and watching the money roll down to them.

Some of this material is taken from Ellenberg's sparkling columns for Slate magazine, and much of it concerns American popular culture. We get baseball statistics, the Bush-Gore election and even dating. Ellenberg uses maths to demonstrate that a single man looking for love should hang around with a friend who is pretty much exactly like you - only slightly less desirable. (Basically, the friend helps you look as good as it is possible for you to look, but of course it’s more complicated than that.)

Despite the strong transatlantic flavour, all the models are designed to show something universal: how the principles of maths can be applied to things that really matter to us. Ellenberg analyses, for example, the trickiness of different electoral systems. Take instant run-off voting (IRV), whereby candidates are listed in order of preference. He manages to show (again, without anything beyond basic arithmetic, though you do have to pay attention) how it is quite possible that every candidate in any given three-candidate election would undoubtedly lose if they faced a one-on-one contest with either of the other two. He also shows how, for the same three-way election, different systems could deliver three different winners. Is there such a thing, in such a case, as a majority choice?

The bigger theme of his book, it becomes clear, is that without an essential grasp of how mathematical systems work we are often at risk of making serious mistakes. Some errors may be relatively harmless. For example, in the mid-1990s millions of readers lapped up the revelations of the Bible code, according to which different arrangements of Hebrew letters (take every fifth letter in Genesis) were said to predict the future. Few probably came to much harm, despite the deficiencies in their statistical awareness exposed by Ellenberg. But how about those investment funds that, year on year, announce dramatic recent track-records? Those returns actually tell you little, Ellenberg explains, unless you also know how many of the broker's other funds did not succeed in this way - and were quietly buried.

And some statistical misunderstandings can have the most dramatic effects on our lives. In 1995, for instance, a scare about a particular kind of oral contraceptive, and the elevated risk of blood clots, caused British women to stop taking the pill in their thousands. The result was some 26,000 conceptions that might not otherwise have occurred. But the root cause of the scare was the statistical language of the warning, which talked about a 'two-fold' increase in the risk ratio. Put another way, it actually meant an increase in probability of some 1 in 7,000.

Underlying the playful stories that make this book so gloriously, surprisingly readable is a passionate argument for the core discipline of managing uncertainty in decision-making. Ellenberg's message is not so much 'how not to be wrong' as 'how to know how wrong you might be'. In short, we dismiss maths at our peril, and this book charmingly, persuasively puts us straight. If only they'd taught maths like this at school.

(Slate article by author just before 2014 midterms in US)

Followers of American politics are converging on a consensus that Republicans are likely to take control of the U.S. Senate following Tuesday's election. But the real political quant nerds, as they refresh their browsers again and again while the returns roll in, will be focusing on a less settled question: How well will Nate Silver's model perform? Will he, as he did in the 2012 presidential election, run the table and get every race right?

Where is the meta-Nate Silver who could make a principled mathematical prediction for how accurate Nate Silver is going to be? Who Nate Silvers the Nate Silvers?

Answer: Nate Silver himself.

Let me explain. As I write this on Monday, the Senate race that Silver's algorithm is most uncertain about is the contest in Kansas, where the editor-in-chief of FiveThirtyEight gives independent Greg Orman a 52 percent chance of winning.* That is, if we ran through this Senate race 100 times - God help us - Silver estimates Orman would win 52 of them. So in those 100 elections, Silver's prediction for Kansas would be correct 52 times and wrong 48 times.

North Carolina isn't quite as close; Silver gives incumbent Democrat Kay Hagan a 71 percent chance of winning. In our imaginary 100-fold Senate race, Silver would get this race right 71 times, but would rack up 29 mistakes. So from Kansas and North Carolina alone, Silver is estimating that he'd make 48 plus 29 equals 77 blown calls. So, as of early Monday afternoon, Silver predicts he'd get 246 predictions wrong if the election were run 100 times, an average of 2.46 per election. This number is what mathematicians call the expected value of the number of wrong predictions. It represents the number of Senate races, on average, Silver expects himself to be wrong about.

At first this might seem a bit contradictory - how can Nate Silver be predicting that Nate Silver is wrong? If you thought you were wrong about something, wouldn't you just .... think the opposite of what you think?

Not really. As a guy who wrote a book called How Not to Be Wrong, I had to become sort of an expert on this. The subtlety is well-captured by an old maxim of the philosopher Willard Van Orman Quine: "[A] reasonable person believes each of his beliefs to be true; yet experience has taught him to expect that some of his beliefs, he knows not which, will turn out to be false. A reasonable person believes, in short, that each of his beliefs is true and that some of them are false."

Or, more succinctly: We always think we're right, but we don't think we're always right. Nate Silver, who has always referred to his perfect accuracy in 2012 as a lucky accident, understands this point very well.

But not everybody does. If you're a die-hard Silver hater, then, the seeming paradox provides an agreeable rhetorical opportunity. If he gets two or three states wrong, as he predicts he will, you can deliver an ex post facto dissection of the obvious political trends in those states that his cold equations failed to capture. And if he gets every state right, you can ding him for underestimating the number of calls he'd miss. Win-win!

That last ding actually isn't that ridiculous. If Silver gets everything right, election after election, it may mean his predictions are improperly calibrated; in that case, when his method says 54 percent, maybe he ought to bump it up to 75 percent, and when it says 75 percent, maybe he ought to call it 90 percent.

But for now, I think Silver's calibration looks pretty good. Somewhat lost in the glow of his bull's-eye in the 2012 presidential election is the fact that he did miscall Senate races that year in Montana and North Dakota. My bet for 2014? Silver is likely to be about as wrong as he expects to be.

Of course, I could be wrong.

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