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Darrell Huff offers an introduction to the theory of probability that you can find in all aspects of life. He weaves in bits of humor and literature to explain the different concepts with real-life examples of coin tosses, card games, roulette, and dice that should help you avoid expensive mistakes.

### The Notes

- “He who has heard the same thing told by 12,000 eye-witnesses has only 12,000 probabilities, which are equal to one strong probability, which is far from certainty.” — Voltaire
- “A reasonable probability is the only certainty.” — E. W. Howe
- “It is truth very certain that, when it is not in our power to determine what is true, we ought to follow what is most probable.” — Rene Descartes
- “Life is a school of probability.” — Walter Bagehot
- “In flouting, or failing to grasp, the laws of chance we hurt ourselves in many ways. We buy insurance we don’t need and we fail to insure the risks we should. We make damaging decisions in business and in driving a car. We flip from overpessimism to unguided optimism and back.”
- 50/50 chances stop being even when viewed in strings.
- To find the probability of getting specific events in a row, multiply together the chance of getting each event. The chance of tossing heads several times in a row? One toss coming up heads is 1/2. Two tosses coming up heads is 1/2 times 1/2, or 1/4. Three tosses of heads are 1/2 times 1/2 times 1/2, or 1/8. And so on.
- The probability of anything occurring falls on a scale of zero to one. Zero is impossible. One is certain. Fractions falling between zero and one are probabilities. All possibilities must add up to one.
- “Most difficulties with probability lie in three areas: inequality of chances; small numbers of cases; and letting history creep in.”
- Inequality of Chances: Always test the assumptions. Loaded dice, double-sided coin, or other deceptive devices (cheating) may be in use if the odds seem
*off*. It changes the probability without you knowing it. - Small Numbers: The results from a small sample size can widely deviate from the average of a large number of occurrences. At the same time, a small number of occurrences is more likely to produce exact odds, while a large sample will produce a better approximation of odds. Example: 1,000 coin tosses will come really close to 50/50 but two, four, or ten tosses might come up all head or all tails, but also likely to produce exactly 50/50.
- History Creep In: The mistaken belief that independent events, strung together, somehow impact the next independent events, like the mistaken idea that ten heads in a row,
*must*mean a tail (or run of tails) is next. It doesn’t. A coin has no memory. “The fallacy — known as “the maturity of the chances” — is the same in every case. Things like cards and roulette wheels have no memory. Their future behavior is not affected by what has occurred in the past. the probability still remains fifty-fifty in each instance, but that is the probability for the future now, not for the total run including the one-sided past.” - The Maturity of Chances = The Gambler’s Fallacy: the mistaken belief that a random event is more or less likely based on a previous series of events. For example: mistakenly believing that a coin that comes up heads ten times in a row is less likely to come up heads an eleventh time.
- The expensive error is “…the attempt to apply to the total of a series, part of which has been completed, the rule that fits only the unknown future.”
- “People have been trying for a long time to control risk, mainly by supplication, sacrifices, and hanging around oracles. These efforts add up to an enormous record of human superstition. Playing a hunch, backing a lucky number, reading predictive value into a coincidence or a dream — each is an attempt to defeat chance. So is reliance on the fallacy of the maturity of the chances.”
- Expectation = (amount you can win) x (the odds of winning)
- If the odds are fair, expectations should equal what’s at stake and you’ll break even. But, in most cases, odds aren’t fair because costs are involved like with insurance, speculation, and roulette where the House takes its cut.
- Insurance vs. Gambling: “In gambling you know the odds or can easily learn them; in insurance they can at least be estimated. In both cases you have a losing expectation, since both kinds of institution must have rent and salary; money and profits. The difference, of course, lies in the nature of the contingency that leads to a payoff. In gambling it is arbitrary: you’re as likely to win when you don’t need the money as when you do. Insurance money, however, comes when have sustained a loss and need it. That’s why insurance is often necessary and roulette is sometimes fun.”
- The Martingale: After each winning bet, you bet the same as the previous bet, but after each losing bet you double the previous bet. It’s subject to sequence risk. A late string of losses could quickly burn through your bankroll and wipe you out.
- The Martingale: “It is mathematically sound in the sense that it has actually permitted you to be almost certain of a small gain. In return for this near-guarantee, however, you have accepted the tiny — but real — risk of a large loss.”
- Basic Principles of Probability: “Every conceivable sequence is as likely as any other.”
- On Probability Theory Models: “It is rather a model of reality — a good model as long as (and where) it is useful, a bad model when extended beyond that point.”
- “Scientific thinkers have given us, and themselves, all sorts of models of the real world. Very useful they can be, too. Euclid’s geometry is one, and Newton’s physics is another. Einstein and others have come along to show us where these models become misleading and we cry in our disillusion that they are false theories. But they are still as “true” (as useful, as workable) as ever where they do fit. And all is well so long as scientists don’t, as artists sometimes do, become too attached to their models.”
- Probabilities are a reminder that “on average” and “for the most part” are results from large sample sizes, not rules of law that small sample sizes are certain to meet.
- On human nature and probability: Our beliefs, superstitions, ignorance, and emotion impact our view on odds.
- The best way to avoid human error and deal with chance: attain maximum information and understand the probability of the situation. Most people fail with this.
- “Gambler’s are inclined to put an equally misplaced faith in what they conceive to be the laws of probability. Any man while gambling is likely to respond primitively. He may believe simultaneously in the contradictory fallacies of catching a run of luck while it is hot and of taking advantage of the ‘maturity of chance.’ In one moment he bets on the number or the player that has won several times in a row. The dice, he says, are hot. Or the player is in the midst of a lucky streak. But, in fact, dice have no memories and luck is of no use in predicting the future but only in describing the past.”
- Beware of claiming luck as skill when you win but put losing down to bad luck.
- “…when I succeeded I raked up my gains, with a half impression that I had been a clever fellow, and had made a judicious stake, just as if I had really moved a skilful move at chess; and that when I failed, I thought to myself, “Ah, I knew all the time I was going wrong in selecting that number, and yet I was fool enough to stick to it,” which was, of course, a pure illusion, for all that I did know the chance was even, or much more than even against me. But this illusion followed me throughout. I had a sense of
*deserving*success when I succeeded, or of having failed through my more wilfulness, or wrong-headed caprice, when I failed.” — Englishman describing his first casino experience in 1873. - “The uncertainties of the world we now ascribe not to the uncertainties of our thoughts, but rather to the character of the world around us. It is a more sensible, more mature and more comprehensible view.” — Charles Darwin
- On survival: A streak of bad luck can ruin a player even in a game where the player has a slight advantage. Having a strategy to ride out bad runs is immensely important.
- “The purpose of game theory is to study strategy. Its frivolous applications is to games, such as poker. Its most serious potential application is to economics — the behavior of men in the marketplace, the strategies of buying and selling, of combinations and monopolies, of speculation and investment.”
- The typical bell curve was once believed to be a naturally recurring result, that scientists treated it like a model. So much so, that they even forced data to fit it. They never considered a skew curve was possible.
- Coincidence, like luck, can explain extremely improbable events, like a series of lucky guesses or predicting the next recession.
- “For short-range forecasts, people notice your misses — but for long-range forecasts, all they notice is your hits.” — Thomas Malone, American Meteorological Society.
- On improbability: a series of highly unlikely events may be practically impossible, but even if it’s a one in a million chance, that
*one*makes it possible. - “Aristotle once remarked that it was probable that the improbable would sometimes take place.”
- “Strange events permit themselves the luxury of occurring.” — Charlie Chan
- “I could only say what was the balance of probability. I did not at all expect to be so accurate.” — Sherlock Holmes
- “Coincidences, in general, are great stumbling-blocks in the way of that class of thinkers who have been educated to know nothing of the theory of probabilities — that theory to which the most glorious objects of human research are indebted for the most glorious illustration.” — C. Auguste Dupin
- On Luck: “We all yearn for the Wonderful, and the Probable is what most of us nearly always get. When someone gets the Wonderful we call him lucky. When he gets the more favorable of even the fairly probable things repeatedly we may still call him lucky. That’s a fair use of the word. But we go wrong when we impute luck to him as something that he possesses, or when we give a predictive value to it.”
- Gamblers often confuse luck with skill, or imbue skill onto lucky streaks, like picking a series of numbers at roulette, despite the odds never changing, always being against them.
- “The law of averages is the law of large numbers. Where chance is an important factor, conclusions based on a few instances remain highly untrustworthy — the fallacy of the small sample.”
- “At least a mild skepticism is in order when the probabilities appear to be flouted… you are entitled to wonder what’s going on if there never seems to be a mistake in your favor.”

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