Improbability Principle by David Hand

The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day

by David J. Hand

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23 Highlights | 17 Notes

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You know very well

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Unfortunately, we always have only approximate knowledge about the present.

See? Forget yesterday and today? We barely know NOW.

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And once the attempt to understand natural physical laws was no longer regarded as blasphemous, it would also no longer be blasphemous to see through chance events to predict likely outcomes,

Authority and social judgment can affect our knowledge and research.

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As we shall see, the fact that the normal distribution is merely an approximation to naturally occurring distributions is very important for the Improbability Principle.

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But Einstein was representative of a small minority. The consensus now is that nature is indeed fundamentally driven by chance—that uncertainty lies at its very core.

Chance all the way down

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We begin with one of the most important strands: the law of inevitability. This is a simple and often overlooked observation, and one that in a real sense underlies everything else: it’s the simple fact that something must happen.

just remember: something or other is going to happen

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Nothing at all is not going to happen. No, instead SOMETHING is going to happen.

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Some mathematicians

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It follows that the chance of winning a lottery twice must surely be astronomically small. But Evelyn Marie Adams won the New Jersey Lottery twice in four months, first in 1985 and then again the next year, collecting a total of $5.4 million.3 The chance of her winning twice in this time span was about one in a trillion.4

I can see the letter: Dear Madam, We are pleased to inform you that you have again won ...

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And that’s just for one lottery. When we take into account the number of lotteries around the world, we see that it would be amazing if draws did not occasionally repeat. So you won’t be surprised to learn that in Israel’s Mifal HaPayis state lottery the numbers drawn on October 16, 2010—13, 14, 26, 32, 33, and 36—were exactly the same as those drawn a few weeks earlier, on September 21. You won’t be surprised to learn that, but scores of people flooded Israeli radio station phone-ins to complain that the lottery was fixed.

Be ready! Some surprising things will happen today and every day.

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Another, rather frustrating, way in which the law of combinations can generate lottery matches is illustrated by what happened to Maureen Wilcox in 1980. She bought tickets containing the winning numbers for both the Massachusetts Lottery and the Rhode Island Lottery. Unfortunately for her, however, her ticket for the Massachusetts Lottery held the winning numbers for the Rhode Island Lottery, and vice versa. If you buy tickets for ten lotteries, you have ten chances of winning. But ten tickets mean 45 pairs of tickets, so the chance that one of the ten tickets will match one of the ten lottery draws is over four times larger than your chance of winning. For obvious reasons, this is not a recipe for obtaining a vast fortune, since matching a ticket for one lottery with the outcome of the draw for another wins you nothing—apart from a suspicion that the universe is making fun of you.

Fate hands out a little frustration !

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Suppose, for example, that we have a class of 30 students. They can interact in various ways. They could work as individuals—there are 30 of them. They could work in pairs—there are 435 different pairs. They could work in triples—there are 4,060 possible different triples. And so on, up to, of course, them all working together—

there is one set of all 30 students working together. In total, the number of different possible groups of students that could be formed is 1,073,741,823. That’s over a billion, all just from 30 students. In general, if a set has n elements, there are 2n–1 possible subsets which could be formed. If n = 100, this gives 2^100–1 ≈ 10^30, a truly large number in anyone’s terms.

Math shows teaching can complicated

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The use of an average here brings to mind the old joke that your average temperature is fine if you have your feet in the oven and your head in the refrigerator.

on average, what is the variation around look like?

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It all adds up to a truly large number of opportunities for holes in one to occur, to the extent that the law of truly large numbers means we should expect to see such events occurring, perhaps with an almost tedious regularity.

expect a hole in one SOMEWHERE

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But careful investigation by the psychologist Oskar Pfungst showed that Hans could get the answer right only if the person asking the question knew what it was: then he was right 89 percent of the time. But if the questioner didn’t know the answer, Clever Hans was right only 6 percent of the time. It turned out that the horse was responding to subliminal cues subconsciously given by the questioner. So: if a horse can do it, what about Hardy’s receivers, responding to subconscious cues from the assistants? Remember, it takes only a tiny shift in the background probabilities to have a large impact on the outcome probabilities.

Some horses are very sensitive to human body language

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Professor Ray Hill of the University of Salford in the UK calculated that “single [SIDS] deaths outweigh homicides by about 17 to 1, double [SIDS] deaths outweigh double homicides by about 9 to 1, and triple [SIDS] deaths outweigh triple homicides by about 2 to 1.”12 There is a factor-of-ten difference between Meadow’s estimate and the estimate based on recognizing that SIDS events in the same family are not independent, and that difference shifts the probability from favoring homicide to favoring SIDS deaths.

Sudden infant death is more frequent than child murder

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Take Major Walter Summerford, who was knocked from his horse by a lightning bolt in Flanders in February 1918, and was temporarily paralyzed from the waist down. After that experience, Summerford moved to

Canada, where he took up fishing—only to have the tree he was sitting under struck by lightning in 1924, paralyzing his right side. He recovered, until he was completely paralyzed by yet another lightning strike in 1930, while walking in the park. He died two years later, in 1932—but not from a lightning strike. But just to rectify the oversight, in 1936 his gravestone was struck by lightning. Clearly life would have been less risky for him if he had taken up knitting.

Pursued by lightning unfairly

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In the same way, we can generalize the birthday problem discussed in chapter 5. While it might be a surprising coincidence if I had the same birthday as you, it would be less of a surprise to find that my birthday was within a week of yours. Indeed, take this far enough by sufficiently relaxing what you mean by “near enough,” and it’s certain that our birthdays will be near each other (I was born on one of the days between January 1 and December 31—and so were you!).

You were born on a day of the year, just like me!

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I wouldn’t have seen it if I hadn’t believed it. —Marshall McLuhan

We do manage to see unexpected things but we look for has an advantage.

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A credit card manager who was not aware of the base rate fallacy might decide to act on the predictions of this instrument. When it flagged a transaction as possibly fraudulent, he might block the card, preventing any further transactions. That’s all well and good, but now suppose I tell you that a ballpark figure is that about 1 in 1,000 credit card transactions is fraudulent. This figure, the 1 in 1,000, is the base rate. Because the number of legitimate transactions is overwhelmingly larger than the number of fraudulent transactions, it’s much more probable that a flagged transaction is actually a misclassified legitimate one than a correctly classified fraudulent one. In fact, the exact probability that the flagged transaction is actually a misclassified legitimate one is 91 percent. That means that despite the fact that the fraud detector labeled 99 percent of the fraudulent transactions correctly and 99 percent of the legitimate transactions correctly, 9 out of 10 of the times when it raised the alarm it was wrong.

Rare events are tricky.

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Suppose we toss

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The behavioral psychologist B. F. Skinner investigated these

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to

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The word “random” here is used in a rather special sense. What it means is that each of the ten digits occurs one tenth of the time, any pair of digits occurs one-hundredth of the time, any triple occurs one-thousandth of the time, etc. Its digits run on forever and never repeat in cycles.