How Not To Be Wrong: The Hidden Maths of Everyday

by Jordan Ellenberg

A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.”

A statistically significant finding gives you a clue, suggesting a promising place to focus your research energy. The significance test is the detective, not the judge.

conditional probabilities; “the probability that X is the case, given that Y is.”

Just as the prior describes your beliefs before you see the evidence, the posterior describes your beliefs afterward. What we’re doing here is called Bayesian inference, because the passage from prior to posterior rests on an old formula in probability called Bayes’s Theorem.

In the Bayesian framework, how much you believe something after you see the evidence depends not just on what the evidence shows, but on how much you believed it to begin with.

The lesson about inference: you have to be careful about the universe of theories you consider. Just as there may be more than one solution to a quadratic equation, there may be multiple theories that give rise to the same observation, and if we don’t consider them all, our inferences may lead us badly astray.

principle of indifference—since there can be no principled way to pretend we don’t know we exist, we just divvy up the prior probability evenly, 50% for GOD and 50% for NO GOD.

“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.”

Here’s how you compute the expected value of a lottery ticket. For each possible outcome, you multiply the chance of that outcome by the ticket’s value given that outcome.

The standard economic story is that human beings, when they’re acting rationally, make decisions that maximize their utility.

We know, at least, that there is no universal curve; different people in different contexts assign different utilities to money. This fact is important. It gives us pause, or it ought to, when we start making generalizations about economic behavior.

Money must not be estimated by its numerical quantity: if the metal, that is merely the sign of wealth, was wealth itself, that is, if the happiness or the benefits that result from wealth were proportional to the quantity of money, men would have reason to estimate it numerically and by its quantity, but it is barely necessary that the benefits that one derives from money are in just proportion with its quantity; a rich man of one hundred thousand ecus income is not ten times happier than the man of only ten thousand ecus; there is more than that what money is, as soon as one passes certain limits it has almost no real value, and cannot increase the well-being of its possessor; a man that discovered a mountain of gold would not be richer than the one that found only one cubic fathom.

In the decision-theory literature, the former kind of unknown is called risk, the latter uncertainty. Risky strategies can be analyzed numerically; uncertain strategies, Ellsberg suggested, were beyond the bounds of formal mathematical analysis, or at least beyond the bounds of the flavor of mathematical analysis beloved at RAND.

variance, a measure of how widely spread out the possible outcomes of a decision are, and how likely one is to encounter the extremes on either end.

“the combinatorial explosion.” Put simply:

very simple operations can change manageably large numbers into absolutely impossible ones.

error-correcting code, a communications protocol that allows the receiver to cancel out the errors in a noisy signal.

the more resistant to noise you want your signal to be, the slower your bits are transmitted.

The Hamming code* is a rule that transforms each of these three-digit blocks into a seven-digit string. Here’s the codebook: 000 -> 0000000 001 -> 0010111 010 -> 0101011 011 -> 0111100 101 -> 1011010 110 -> 1100110 100 -> 1001101 111 -> 1110001

geometry, the Hamming code has the same magical error-correcting property as “repeat three times”; if a message gets modified by a single bit en route, the receiver can always figure out what message the transmitter meant to send.

a completely random code, with no design at all, was extremely likely to be an error-correcting code.

the fact that no two tickets agree on five out of six numbers means that this code, like the Hamming code, has no two code words separated by a Hamming distance of less than four.*

In the Kahnemann-Twersky theory, people tend to place more weight on low-probability events than a person obedient to the von Neumann-Morgenstern axioms would; so the allure of the jackpot exceeds what a strict expected utility calculation would license.

Complete freedom to enter trade and the continuance of competition mean the perpetuation of mediocrity. New firms are recruited from the relatively “unfit”—at least from the inexperienced. If some succeed, they must meet the competitive practices of the class, the market, to which they belong. Superior judgment, merchandising sense, and honesty, however, are always at the mercy of the unscrupulous, the unwise, the misinformed and the injudicious. The results are that retail trade is over-crowded, shops are small and inefficient, volume of business inadequate, expenses relatively high, and profits small. So long as the field of activity is freely entered, and it is; and so long as competition is “free,” and within the limits suggested above, it is; neither superiority nor inferiority will tend to persist. Rather, mediocrity tends to become the rule. The average level of the intelligence of those conducting business holds sway, and the practices common to such trade mentality become the rule.

Excellence doesn’t persist; time passes, and mediocrity asserts itself.

Tall people are tall because their heredity predisposes them to be tall, or because external forces encourage them to be tall, or both. And the taller a person is, the likelier it is that both factors are pointing in the upward direction.

We say that the conditional expectation of the son’s height (that is, how tall the son will be on average given that his father stands six foot one) is the same as the unconditional expectation (the average height of sons computed without any restriction on the father).

curves on his scatterplot along which the density of points was roughly constant. Curves of this kind are called isopleths,

eccentricity of the ellipse, is a measure of the extent to which the height of the father determines that of the son. High eccentricity means that heredity is powerful and regression to the mean is weak; low eccentricity means the opposite, that regression to the mean holds sway. Galton called his measure correlation, the term we still use today.

compression, the critical mathematical technology that allows images, videos, music, and text to be stored in much smaller spaces than you’d think. The presence of correlation makes compression possible;

Darwin showed that one could meaningfully talk about progress without any need to invoke purpose. Galton showed that one could meaningfully talk about association without any need to invoke underlying cause.

In geometry, we call a pair of vectors that form a right angle perpendicular, or orthogonal.

Niacin is correlated with high HDL, and high HDL is correlated with low risk of heart attack, but that doesn’t mean that niacin prevents heart attacks.

Undecided voters, by and large, aren’t undecided because they’re carefully weighing the merits of each candidate, unprejudiced by political dogma. They’re undecided because they’re barely paying attention.

Doll and Hill’s data showed that lung cancer and smoking were correlated; their relation was not one of strict determination (some heavy smokers don’t get lung cancer, while some nonsmokers do), but neither were the two phenomena independent. Their relation lay in that fuzzy, intermediate zone that Galton and Pearson had been the first to map.

But one thing’s for certain: refraining from making recommendations at all, on the grounds that they might be wrong, is a losing strategy. It’s a lot like George Stigler’s advice about missing planes. If you never give advice until you’re sure it’s right, you’re not giving enough advice.

Each voter has a perfectly rational, coherent political stance. But in the aggregate, their position is nonsensical.

If there’s no such thing as the public opinion, what’s an elected official to do? The simplest answer: when there’s no coherent message from the people, do whatever you want. As we’ve seen, simple logic demands that you’ll sometimes be acting contrary to the will of the majority. If you’re a mediocre politician, this is where you point out that the polling data contradicts itself. If you’re a good politician, this is where you say, “I was elected to lead—not to watch the polls.”

“independence of irrelevant alternatives.”

if you have a choice between two options A and B, the presence of a third option C shouldn’t affect which of A and B you like better.

“asymmetric domination effect,”

So if you’re a single guy looking for love, and you’re deciding which friend to bring out on the town with you, choose the one who’s pretty much exactly like you—only slightly less desirable.

the apparent irrationality of popular opinion can arise from the collective behavior of perfectly rational individual people.

The appeal of instant-runoff voting (or “preferential voting,” as they call it in Australia) is obvious.

Condorcet paradoxes,

suppose a seven-person jury has to decide a defendant’s guilt. Four say the defendant is guilty, and only three believe he’s innocent. Let’s say each of these citizens has a 51% chance of holding the correct view. In that case, you might expect a 4–3 majority in the correct direction to be more likely than a 4–3 majority favoring the incorrect choice.

If the majority of people believe something, Condorcet said, that must be taken as strong evidence that it is correct.

“I must act not by what I think reasonable,” Condorcet wrote, “but by what all who, like me, have abstracted from their own opinion must regard as conforming to reason and truth.”

If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.

formalism in the law manifests itself as an adherence to procedure and the words of the statutes, even when—or especially when—they cut against what common sense prescribes.