Self Help

The Drunkard's Walk - Leonard Mlodinow

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Matheus Puppe

· 46 min read



Here is a summary of the key points from the contents section:

  • The book explores the principles that govern chance and randomness, how they play out in human affairs like politics, business, medicine, and sports, and how people often make mistakes in understanding and interpreting random situations.

  • It discusses cognitive biases and flawed thought processes that people exhibit when dealing with uncertainty or imperfect information.

  • The chapters cover topics like probability, counting possible outcomes, laws of large and small numbers, conditional probability and medical testing, measurement error, patterns and illusions in random data, and how chance is a more fundamental concept than causality.

  • The prologue shares an anecdote about the author’s father talking to him as a teenager about chance events and randomness, using his own experience of surviving the Holocaust as an example of how seemingly random circumstances can profoundly impact one’s life.

  • Chapter 1 starts recounting the author’s childhood memory of watching randomly flickering candle flames and his father explaining that life isn’t always predictable and sometimes things happen that cannot be foreseen.

  • Daniel Kahneman won the Nobel Prize in Economics for his work studying cognitive biases and misperceptions of randomness with Amos Tversky. Their research was sparked by a random event - an encounter with Israeli air force flight instructors.

  • The instructors believed that praising good maneuvers led to worse future performance, while screaming at bad maneuvers led to improvement. Kahneman realized this was due to regression to the mean - extraordinary performances tend to be followed by more average ones purely due to chance.

  • Any especially good or poor performance was mostly a matter of luck, not real changes in ability from one maneuver to the next. So a good performance would likely be followed by a average one, making praise seem ineffective. And a poor performance would likely be followed by an average one, making criticism seem effective.

  • Kahneman’s insight led him to study cognitive biases and how people misperceive randomness. His work with Tversky clarified common fallacies in understanding chance events and uncertainty. Their research helped explain why intuitive thinking can lead to less than optimal decisions when assessing risk and uncertainty.

This passage discusses the research of psychologists Daniel Kahneman and Amos Tversky on human intuition and decision-making under uncertainty. Some key points:

  • Kahneman was intrigued by the intuitive belief of flight instructors that harsh criticism improved student performance, when research showed it made no difference.

  • He and Tversky found that even sophisticated people’s beliefs and intuitions about probabilities and random processes (e.g. in business, sports) often let them down.

  • Acceptance or rejection of books, movies, etc. involves a lot of randomness and uncertainty. Many hugely successful works faced repeated rejections first.

  • Hollywood success in particular is very hard to predict in advance due to many uncontrollable factors. Box office outcomes are often erroneously attributed to skill when luck plays a bigger role.

  • In general, we underestimate the effects of randomness on outcomes and overestimate how directly results reflect skill or actions. Kahneman and Tversky’s research showed intuitions about uncertainty can be systematically mistaken.

  • The passage discusses randomness and luck in Hollywood and uses the example of Sherry Lansing, who led Paramount Pictures to great success but was suddenly fired after a few years of box office underperformance.

  • It argues Lansing’s downfall was largely due to randomness and bad luck, not actual mistakes in her leadership. Films released after she left continued Paramount’s success, indicating her previous strategies were still working.

  • More broadly, it discusses how the clustering of random events can easily be misinterpreted, leading people to wrongly ascribe successes and failures to skill or lack thereof. Outcomes depend significantly on luck.

  • The author had a similar realization about randomness in sports while studying probability in college. He analyzes the 1961 home run race between Roger Maris and Mickey Mantle as an example, where Maris’ single record-breaking season was dismissed despite his overall performance being strong with an element of luck.

So in summary, the passage uses examples from Hollywood and sports to illustrate how randomness and luck strongly influence outcomes but are often misattributed to skill or lack thereof due to poor understanding of probabilities and the clustering of random events.

  • The passage describes an experiment conducted by Daniel Kahneman and Amos Tversky where they presented subjects with a description of a woman named Linda and asked them to rank 8 statements about Linda’s possible occupations in order of probability.

  • Most subjects ranked “Linda is active in the feminist movement” as the most probable, which aligns with her described attributes and interests from college.

  • However, subjects also ranked “Linda is a bank teller and is active in the feminist movement” as more probable than the simple statement “Linda is a bank teller.”

  • This result violates the logical rule that a specific statement cannot be more probable than a more general/inclusive one. For example, the probability of Linda being a bank teller AND feminist cannot be higher than just the probability of her being a bank teller.

  • The experiment revealed people’s intuitions about probability do not always align with the basic logical rules, indicating probabilistic reasoning is subtle and understanding can be improved through careful thinking and experience with these concepts.

So in summary, the Kahneman/Tversky experiment showed how people’s intuitions can violate basic probability laws and highlighted the challenges of properly understanding and applying probabilistic reasoning.

  • Kahneman and Tversky conducted an experiment where they presented descriptions of a woman named Linda and asked people to rank the probabilities of different statements about her.

  • Most people ranked “Linda is a bank teller and active in the feminist movement” as more probable than “Linda is a bank teller”. This violates the law of probability, as the probability of two events occurring together cannot be greater than the probability of one occurring individually.

  • Even when explicitly told about this law of probability, some people still ranked the joint probability higher. Kahneman and Tversky concluded this was due to the detail “active in feminist movement” making the scenario seem more plausible, even though it actually makes it less probable.

  • They found this “conjunction fallacy” occurs because people judge probabilities based on how well a story or scenario “fits” their mental model, rather than strictly following logic.

  • Doctors and lawyers were also found to commit this error by assigning higher probabilities to scenarios involving multiple details rather than single events.

  • While illogical, this cognitive bias may have evolved because mistakenly seeing patterns is less costly than missing potential opportunities or threats in our environment.

So in summary, Kahneman and Tversky discovered people systematically violate the rules of probability judgment by favoring more detailed scenarios, even when they are logically less probable - a cognitive bias they termed the “conjunction fallacy”.

Here are the key points about why the Greeks did not develop a theory of probability:

  • Many Greeks believed that future events were determined by the will of the gods, not random processes. So understanding randomness was not seen as important or possible.

  • Greek philosophy emphasized absolute truth through logic and axioms, not uncertain claims. Philosophers like Plato frowned upon arguments based on probabilities and likelihoods.

  • Record keeping was limited, making it difficult to systematically study the frequencies of past events and estimate probabilities. Memory biases like availability bias distorted perceptions.

  • The Greeks lacked a usable number system for arithmetic calculations. Their alphabetic number system was awkward and they had no concept of zero. Probability requires arithmetic, which was not fully developed until later.

  • It was not until the development of modern number systems and symbols for arithmetic operations that the foundations were in place for a theory of probability to emerge in later centuries. The Romans were more practical but still did not produce a mathematical theory of probability.

  • The passage discusses the origins and early development of probability theory, starting with Cicero in ancient Rome. Probabilis, meaning “probable”, comes from Cicero.

  • Roman law started incorporating mathematical thinking and probability concepts to deal with conflicting evidence and testimonies. They developed ideas like “half proofs” where evidence was neither fully believed or disbelieved.

  • However, the Romans did not fully grasp compound probability. They incorrectly added probabilities rather than multiplying them. For example, two “half proofs” would be less than a full proof according to correct probability.

  • The passage then outlines three main laws of probability: 1) Independent events are multiplied. 2) Mutually exclusive events are added. 3) The sum of all outcomes is 1.

  • It notes the Romans added probabilities incorrectly. And our modern legal system still has room for improvement, like not fully presenting error rates in forensic evidence like DNA testing. In the end, probability theory has its origins in trying to solve problems with uncertain evidence and reasoning in ancient Roman law.

  • Timothy Durham was sentenced to over 3,100 years in prison for rape in Oklahoma, even though 11 witnesses placed him in another state at the time.

  • Initial DNA analysis by the lab failed to fully separate the rapist and victim’s DNA in a tested fluid sample, combining their DNA and producing a match to Durham.

  • A retest found this error, and Durham was released after nearly 4 years in prison.

  • Estimates of error rates in DNA analysis vary, but many experts say around 1%. However, courts often don’t allow testimony on overall error rates.

  • Jurors may assume the overall error rate is between the rare accidental match rate (1 in 1 billion) and common lab error rate (1 in 100), but using probability, the actual rate is closer to the more common lab error rate of 1 in 100.

  • This calls into question claims of DNA being infallible, as the higher lab error rate is often not disclosed in court.

  • One example is People v. Collins in 1968, where witnesses couldn’t identify the suspects but a math instructor used the product rule on independent probabilities of characteristics to argue the chances suspects were innocent were 1 in 12 million.

  • However, the characteristics weren’t independent, and the relevant probability was the chances the matching suspects were guilty, not randomly selected individuals matching the description.

  • The supreme court overturned due to these probability errors, but courts still grapple with properly applying statistics and probability in legal cases.

  • Gerolamo Cardano (1501-1576) was an Italian polymath who published over 130 books on various topics including philosophy, medicine, mathematics and science.

  • In his later years he fell into poverty and obscurity. In 1576, just before his death, he burned around 170 unpublished manuscripts. 111 survived, including his groundbreaking work on probability and games of chance titled “The Book on Games of Chance”.

  • This was the first scientific text on probability theory and randomness. It introduced the concept of analyzing all possible outcomes of uncertain events, known as the “sample space” approach, which formed the basis for mathematical probability.

  • However, Cardano himself still believed in fate and divination. His understanding of probability was more intuitive from studying games, rather than from a strictly rational scientific viewpoint limited by the mathematics of his time.

  • Nonetheless, his work on analyzing random processes through considering all outcomes was pioneering and set the foundation for centuries of mathematical probability theory to come. It demonstrated how chance situations could be systematically approached using simple analytical methods.

  • In the early 16th century, algebra and arithmetic were still in their infancy, preceding even the invention of the equal sign. Gebra and arithmetic were not advanced mathematical tools at that time.

  • Gerolamo Cardano, born in 1501, helped advance the understanding and use of probability. He wrote one of the earliest books on probability and games of chance called Book on Games of Chance.

  • Living when he did, Cardano had the advantage of knowledge that had been developed by Hindus and Arabs, including the use of positional notation in base 10 and progress in the arithmetic of fractions, both crucial for probability analysis.

  • In Book on Games of Chance, Cardano considered random processes where each outcome was equally likely, like the roll of dice. He formulated an early version of the concept of a sample space and stated a “general rule” about calculating probabilities that represented an important stepping stone.

  • While not perfect, Cardano’s work established an early beachhead in the human quest to understand the nature of uncertainty and probability through analyzing games of chance. It helped advance mathematical tools and concepts that were still in their infancy in his time.

  • In the 18th century, the French mathematician Jean d’Alembert misapplied probability concepts when analyzing the toss of two coins. He incorrectly concluded the chance of each outcome (0, 1, or 2 heads) was 1/3.

  • One of Cardano’s advances was systematically analyzing sequences of events like coin tosses. The key is considering all possible sequences (heads-heads, heads-tails, etc.), not just total outcomes.

  • For two coin tosses, the sample space is the 4 possible sequences. Cardano showed the chances of 0 or 2 heads is 1/4, while 1 head is 1/2 - correcting d’Alembert.

  • The “two daughters problem” is mathematically identical if girls=heads and boys=tails. This “isomorphism” saves work by applying the coin toss solution.

  • Additional questions about conditional probabilities, like the chances of two girls given one is known to be a girl, also require carefully considering the full sample space and eliminating possibilities.

So in summary, it outlines d’Alembert’s incorrect probability application, Cardano’s improvement in systematically considering all outcomes as sequences, and how this approach solves related problems like the two daughters problem.

  • Gerolamo Cardano was a 15th/16th century Italian mathematician, physician and gambler who wrote one of the earliest works on probability and games of chance called Book on Games of Chance.

  • In his time, mathematical and probability theories were not well understood or appreciated. Superstitions and mystical beliefs held more weight than rational analysis. So Cardano’s work had little impact and was not published for over 100 years after he wrote it.

  • Had he lived a few decades later during the Scientific Revolution, Cardano’s work may have been better received. The Scientific Revolution challenged old ways of thinking based on mysticism and embraced systematic observation and mathematical description of natural phenomena.

  • Factors like the lack of advanced algebraic notation at the time and prevailing mystical beliefs hindered appreciation of Cardano’s early contributions to probability and gambling theory. But developments in rational thought during the Scientific Revolution could have made his work more influential if published later.

  • Overall, the timing of Cardano’s work and prevailing intellectual climate help explain why his probability writings had little impact during his own lifetime but became more influential once published after rationalism spread during the Scientific Revolution.

  • Galileo made important early observations on randomness and probability through studying the swinging of a pendulum and analyzing dice games. This represented a shift toward empirical observation and experimentation over intuition.

  • He solved a problem for his patron about why rolling three dice is more likely to result in a total of 10 rather than 9. He implicitly used the principle that probability depends on the number of ways an outcome can occur.

  • This led to further developing methods for systematically analyzing how many combinations or ways there are for events to happen, known as counting principles.

  • Examples are given where not properly accounting for the number of combinations can lead to unexpected outcomes, like a lottery incorrectly having the same number chosen twice when drawing 500 numbers from a pool of 2.4 million.

  • The text discusses how these early foundations for understanding randomness and probability were crucial to later advances, though Galileo seemed to view his dice work as just a commissioned task rather than a new approach. Someone from the next generation would take probability counting principles to new heights.

  • Pascal became interested in gambling and probability after being advised by doctors to divert his mind from studying to relax and socialize due to health issues.

  • Through a friend, Pascal met a gambler named Chevalier de Méré who posed an unsolved probability problem to him called the “problem of points.”

  • This involved calculating the fair odds of dividing a betting pot if a game is interrupted partway through with one player in the lead.

  • Pascal realized solving this problem required new mathematical methods and collaborated with Pierre de Fermat via correspondence to develop solutions.

  • They each independently came up with approaches to calculate probabilities of different outcomes in interrupted games based on considering all possible sequences of wins and losses.

  • Pascal’s method was found to be simpler and more generalizable. This laid the foundations for the modern theory of probability through examining chance events.

  • The collaboration between Pascal and Fermat is considered one of the great correspondences in the history of mathematics for its impact on developing probability theory.

  • Pascal and Fermat proposed that if the Yankees-Braves World Series had ended after 2 games, with the Braves up 2-0, the odds of each team winning the series should be based on calculating all possible outcomes over the remaining games. This gave the Braves an 81% chance and Yankees a 19% chance.

  • The same reasoning could be applied before any games are played, weighting each outcome by its relative probability based on how often each team is expected to win individual games. This shows that the inferior team can still win a 7-game series about 4 times out of 10 if the better team’s odds are 55-45.

  • Pascal’s triangle provides a systematic way to count outcomes without explicitly listing them, which is needed for larger numbers. It allows calculating the number of ways to select objects from a group.

  • Examples are given showing how Pascal’s triangle can be used to understand probabilities in situations like focus groups, where small sample sizes may not accurately reflect the overall population. Agreement in small groups is often random rather than statistically significant.

  • In summary, Pascal and Fermat introduced logical probability analysis of outcomes, and Pascal developed his triangle as a tool to systematically calculate probabilities, even for more complex scenarios.

  • In 1662, servants found writings hidden in Pascal’s jacket describing a mystical experience he had in 1654 where God came to him during a two hour trance.

  • After this experience, Pascal drastically changed his lifestyle, giving up wealth and friends to focus on religion. He still continued writing works like Pensées.

  • Within Pensées, Pascal unexpectedly included a mathematical analysis weighing the pros and cons of believing in God using probability calculations. This became known as Pascal’s Wager.

  • Pascal’s Wager introduced the important concept of expected value/mathematical expectation to decision making under uncertainty. It is considered a founding work of game theory.

  • The passage then provides various examples using mathematical expectation to analyze scenarios like lotteries, sweepstakes and parking meters to demonstrate how thinking in terms of probabilities can reveal unexpected results.

  • It describes an ambitious scheme by Australian investors who analyzed the expected value of buying every possible lottery ticket combination, finding it would be profitable given the lottery jackpot and odds. Their elaborate plan to purchase the tickets mostly succeeded.

  • A consortium in Italy formed to purchase lottery tickets for a large jackpot prize. They divided the work of printing tickets among stores and hired couriers to collect them. However, they ran out of time and only purchased 5 million tickets out of the total 7 million tickets.

  • After the winning ticket was announced, no one claimed the prize for several days. It then took the consortium members time to find the winning ticket among the tickets they purchased. When lottery officials discovered what the consortium had done, they initially refused to pay out the prize but eventually did after a month of legal issues.

  • Pascal made important contributions to the study of randomness through his ideas about counting and the concept of mathematical expectation. However, his health deteriorated and he died at age 39 from a brain hemorrhage after years of illnesses. An autopsy also found lesions in his liver, stomach and intestines.

The passage discusses different interpretations of randomness - the frequency interpretation and subjective interpretation. It also explains Jagger’s exploitation of an imperfection in a roulette wheel at Monte Carlo casino in 1873.

The frequency interpretation judges randomness based on how a sample turns out. The subjective interpretation judges based on how the sample is produced. A throw of a die would be random in theory but not practice under these definitions.

The Rand corporation attempted to generate truly random numbers but found biases, like imperfect dice. They published the numbers despite imperfections.

Jagger noticed some roulette wheel numbers came up more frequently due to imperfections. He bet on these numbers and gained $300,000 before the casino responded by switching wheels. Eventually the casino moved the frets each night, nullifying Jagger’s advantage. He left with $325,000.

Even perfectly balanced systems won’t produce equal frequencies by chance. This raised questions about sampling probabilities that were answered through later mathematical revolutions involving calculus. Jagger’s success relied on inherent imperfections but wasn’t a guaranteed success either.

  • In 1681, Jakob Bernoulli published a pamphlet claiming that comets follow natural laws rather than God’s will. This went against theological views of the time.

  • Bernoulli succeeded Peter Megerlin as professor of mathematics at the University of Basel in 1686. He was interested in problems involving probability and games of chance.

  • Bernoulli was influenced by Christiaan Huygens’ work on probability, but saw limitations in only considering games of chance. He believed probabilities could be determined through observation as well.

  • In 1686, Gottfried Leibniz published his work laying out integral calculus. Isaac Newton also developed calculus but published later. Their work provided tools for Bernoulli’s research on probability.

  • Calculus concepts like sequences, series, and limits were important to Bernoulli. He was one of the first to formally treat how observed frequencies reflect underlying probabilities with increasing trials.

So in summary, Bernoulli made advances in studying probability through a mathematical approach involving observation and analysis of sequences/series, influenced by emerging calculus concepts from Newton and Leibniz. He helped lay foundations for modern probability theory.

  • Zeno’s paradox concerned the amount of time it takes to travel a distance, not the distance itself. If movement is continuous rather than occurring in discrete intervals, then travel is possible in a finite amount of time even if the distance is divided infinitely.

  • Bernoulli investigated what happens when repeated observations or trials are taken, specifically looking at the limit as the number of trials approaches infinity. He found that as more trials are taken, the observed outcomes converge on the true underlying probabilities.

  • Bernoulli formulated this idea mathematically as his “golden theorem,” also known as the law of large numbers. It states that by taking a sufficiently large number of random trials or observations, one can be highly confident that the observed outcomes will be very close to the true probabilities, within any specified tolerance or error.

  • Bernoulli used examples involving drawing colored pebbles from an urn to illustrate the theorem. It applies more generally to any random process with two possible outcomes, like coin flips.

  • The theorem has two parts - stating that the number of trials needed is finite, and providing a formula to calculate that number. The formula was impractical but the core conceptual idea remains valid.

So in summary, Bernoulli used repeated random trials and the concept of limits to show that observed frequencies converge on true probabilities as more observations are taken, addressing Zeno’s paradox of infinitely divided distances.

  • Bernoulli established a very demanding standard of “moral certainty” requiring success more than 99.9% of the time, while today we consider statistical significance to be less than a 5% chance of being wrong.

  • With samples of 370-1000 people, statisticians can accurately estimate population percentages within a margin of error of 2-5%. However, Bernoulli’s goal of near-perfect accuracy requires much larger samples.

  • The “law of small numbers” is a mistaken belief that small samples accurately reflect underlying probabilities. Samples need to be large enough for the law of large numbers to apply.

  • Even with a known 60% probability of success, the chances a CEO will have exactly 3 successful years out of 5 is only 1 in 3, showing small sample results are often unrepresentative.

  • The gambler’s fallacy is the mistaken belief that past random outcomes affect future probabilities - e.g. thinking a coin is “due” after many heads. Probability remains the same with each independent trial.

  • Bernoulli’s manuscript was unfinished at his death. His nephew edited what was completed but felt unqualified for the full task, so applications were never fully developed. Rival mathematician Johann Bernoulli, known for dishonesty, refused the work, possibly to prevent his late brother’s ideas from being disseminated.

  • A Harvard psychology professor had a student who believed he was the subject of an elaborate secret experiment led by B.F. Skinner. The student’s theory was that strange coincidences in his life were part of this experiment.

  • The student was later sued by his former employer and a psychiatrist testified he had a paranoid delusion, citing the student’s claim about an 18th century minister named Thomas Bayes who created a theory of probability.

  • The professor confirmed that Thomas Bayes was a real minister who invented the theory of conditional probability, showing how probabilities change given new evidence or conditions.

  • The theory of conditional probability is what Bayesian reasoning is based on. It involves assessing the probability of an event given or if other events occur.

  • The student’s calculations may have been dubious but the psychiatrist was wrong to dismiss Bayes and his theory of conditional probability out of ignorance. This story illustrates how ignorance of Bayesian reasoning can lead to mistakes in diagnosis and legal judgment.

  • Bayesian reasoning is part of everyday life, like assessing the probability that your boss is responding slowly to emails because your career is falling versus them just being busy. Or a wife incorrectly thinking her husband’s secrecy means he’s cheating rather than dancing.

  • The passage introduces a variant of the two-daughter problem where one child is revealed to be a girl named Florida.

  • It asks whether the chances of two girls is still 1/3 as in the original problem.

  • The author states that the answer is no, and the chance is actually 1/2. However, the reasoning is not yet provided.

  • The passage then briefly discusses Thomas Bayes and how he developed the concept of conditional probability to infer probabilities from observations.

  • It provides an example of how Bayesian analysis is used by insurance companies to determine risk levels and premiums for new drivers based on prior probabilities and new data over time.

  • The author says they will now apply Bayes’ approach of pruning the sample space based on new information to solve the girl-named-Florida problem and show why the chances are 1/2, not 1/3.

So in summary, it sets up the variant problem, states the chances are different than the original problem but doesn’t yet explain why, and provides background on Bayesian probability before promising to resolve the variance using those principles.

  • The text describes a scenario where the speaker is told by their doctor that they have a 999/1000 chance of dying within a decade after testing positive for HIV.

  • However, the doctor misinterpreted the statistics. The 1/1000 number referred to the false positive rate of the HIV test, not the speaker’s actual probability of being HIV positive given a positive test result.

  • Using Bayes’ theorem, the text walks through calculating the correct probabilities. Assuming a 1% prevalence of HIV in the relevant population (heterosexual white males) and a 1/1000 false positive rate:

    • Of 10,000 tests, roughly 1 person would test positive due to true infection
    • About 10 people would test positive due to false positives
  • Therefore, the chance of actually being infected given a positive test is 1/11, not 1/1000 as the doctor stated.

  • The text notes the importance of considering prevalence - if the speaker was from a high risk group like gay men, the post-test probability would have been much higher given the higher prevalence.

  • In summary, the doctor made the common mistake of confusing test accuracy rates with actual post-test probabilities, failing to properly apply Bayesian reasoning. This can have serious consequences when communicating health risks.

  • The passage discusses the prosecutor’s fallacy, which is incorrectly believing that the probability of a certain test result equals the probability that someone is guilty.

  • It gives the example of Mary Decker Slaney, who was accused of doping based on a drug test result. However, taking conditional probabilities into account, the test result only meant an 84.7% chance she was guilty, not 99% as many believed.

  • Sally Clark was wrongly imprisoned for murdering her two infants based on the prosecutor claiming the odds of both dying of SIDS was 73 million to 1. But this did not account for conditional probabilities - the relevant statistic was the relative likelihood of SIDS deaths versus murder.

  • Alan Dershowitz employed the prosecutor’s fallacy to help defend OJ Simpson, focusing on the low probability that a man who batters his wife will murder her, rather than the high probability that a murdered battered wife was killed by her abuser.

  • These cases illustrate how failing to properly apply conditional probability can lead to incorrect assessments of guilt and unjust convictions. Understanding conditional probabilities is crucial to avoid fallacious reasoning in legal contexts.

  • The passage discusses the distinction between probability and statistics. Probability deals with predictions based on known probabilities, while statistics deals with inferring probabilities from observed data.

  • It focuses on the work of Laplace, who developed statistical methods without awareness of Bayes’ theorem. Laplace sought to determine the true value of a measured quantity given a series of measurements, and the likelihood estimates would be close to the true value.

  • His political adaptability allowed him to continue his groundbreaking statistical work through turbulent times in France. Ultimately his analysis was more complete than Bayes’.

  • The passage then shifts to discussing measurement error and the normal distribution/bell curve. Grades, votes, and other measurements are prone to random error and variability. Close elections may see multiple recounts that change results due to this inherent imprecision.

  • Taking averages helps reconcile discordant measurements, though measurements will always have some uncertainty. Statistics provides tools to quantify this uncertainty from observations of real-world data.

  • In the 18th century and earlier, scientists would often report a single “golden number” measurement rather than averages or ranges, considering variation to be a sign of failure.

  • Developing accurate theories of planetary motion required reconciling complex mathematics with imperfect observations and measurements.

  • In the late 18th century, a new rigorous tradition of experimental physics arose in France led by figures like Pierre-Simon Laplace and Antoine Lavoisier. This mathematized the field.

  • Their work, along with Coulomb’s experiments, helped develop the metric system of standardized units to replace disparate existing systems.

  • Understanding random error in measurements became a key task, giving rise to the new field of mathematical statistics and tools for interpreting scientific data and addressing real-world issues.

  • However, uncertainty in measurements is often overlooked when results are reported. Even small changes may not indicate real changes given natural variation.

  • Subjective measurements like essay or wine ratings also show significant inconsistencies between raters, but are still treated as highly precise. Understanding measurement uncertainty is important both for science and everyday life.

Here are the key points from the passage:

  • Wine ratings are questionable because taste perception depends on both taste buds and smell, and it’s difficult to identify flavors in complex mixtures.

  • Expectations also affect taste perception. Studies have shown people perceive wines as tasting sweeter or more expensive if they believe that to be the case.

  • When presented scents out of context, even experts have difficulty identifying them correctly.

  • In wine tasting experiments where tasters had to identify samples or rank wines based on attributes, experts performed only slightly better than chance.

  • While rating systems are imperfect, wine critics continue to use numerical ratings because consumers find them more convincing than vague descriptions.

  • For measurements to have meaning, variability in the data needs to be understood. Concepts like average, standard deviation, and error distribution are important to determine true values from a series of measurements.

So in summary, the passage questions the objectivity of wine ratings due to factors like subjective taste perception and expectations. It cites studies showing experts have limited ability to objectively evaluate wines. And it discusses the importance of variability and error analysis for interpreting measurement data.

This passage summarizes several key points about mathematical statistics and the normal distribution:

  • Mathematical statistics is a coherent subject because the distribution of errors follows predictable patterns, even when measurements have different goals (e.g. positioning of Jupiter vs weight of bread).

  • The normal distribution often describes how data is distributed, even when other factors could influence results (e.g. wine ratings influenced by red vs white wine preferences).

  • Abraham De Moivre discovered the bell curve approximation in 1733 while studying Pascal’s triangle. The bell curve, later called the normal distribution, provides a better approximation than prior methods.

  • The normal distribution describes how most observations fall around the mean, with fewer observations deviating further from the mean in a symmetrical pattern. It is characterized by its mean and standard deviation.

  • Polling and sampling data typically follows the normal distribution, with the margin of error describing the range where results are expected to fall 95% of the time. Sample size affects margin of error.

  • Random processes like coin toss guessing produce results that follow the normal distribution, demonstrating its wide applicability in modeling random variation in measurements and estimates.

  • The passage discusses how political polls reporting approval ratings for a president will often have margins of error of over 5%, which is considered unacceptable by standards. Yet we often make judgments based on just a few data points like individual polls.

  • It notes that after a 2004 Republican National Convention, a CNN poll found President Bush’s approval rating rose 2 percentage points, but neglects to mention the poll’s margin of error was 3.5 points, making the reported change meaningless.

  • The concept of the normal distribution and bell curve is introduced to explain variation in data and measurement error. Multiple independent surveys on the president’s approval rating were given as an example where the spread of results was more likely than exact agreement.

  • Gaussian mathematician Carl Gauss first proposed the normal distribution described measurement error but his proof was invalid. Laplace later realized Gauss’ work could be used to improve his own, building a better case for the normal distribution as the error law.

  • The central limit theorem is discussed as explaining why the normal distribution accurately models error - the summation of many small, independent random factors will follow a normal distribution.

  • In summary, the passage examines how statistical concepts like the normal distribution, margins of error, and central limit theorem provide important context for interpreting individual data points and survey results.

The passage discusses how statistical analysis of large populations can reveal orderly and predictable patterns, even when individuals appear to act randomly. It analyzes historical examples like yearly US driving distances and fatalities that remain consistent in the aggregate, despite people driving different amounts each year.

It then provides background on the early history of statistics. The first national census was conducted in 1086 by William the Conqueror to inventory England’s land and resources for taxation. Early mortality records in London from the 1600s were analyzed by John Graunt, considered a founder of statistics. He drew insights about issues like starvation rates and theories of plague transmission. His work showed how statistics can provide understanding of the systems they represent.

Graunt’s friend William Petty also employed early statistical reasoning to analyze national issues from the perspective of maximizing the sovereign’s interests. His analysis treated members of society as objects that could be manipulated. Overall, the passage discusses how examining large-scale population data revealed order and predictability despite individual variability, laying foundations for the emergence of statistics as a field of study.

  • William Petty advocated forcibly relocating most Irish people to England to increase the wealth of the English kingdom. However, Petty himself owed his own wealth to taking advantage of the invasion of Ireland by stealing property.

  • John Graunt’s work estimating London’s population through analysis of birth and mortality records helped establish the field of statistics. He published one of the first life tables showing life expectancy rates. His work showed populations recover quickly from epidemics through immigration and birth rates.

  • Adolphe Quételet further advanced the field by showing how social phenomena like crime, marriage, and suicide clustered around statistical averages and normal distributions. He found statistical patterns even in deviations, like an excess of short men drafting into the military.

  • Quételet’s work established that statistical analysis could be used to detect irregularities and even catch wrongdoers, as when Jules-Henri Poincaré used distributions to show a baker was shortchanging customers on bread weights. This led to the modern field of forensic economics applying statistics to detect fraudulent behavior in large datasets.

  • Economist Justin Wolfers studied point spreads for college basketball games set by Las Vegas bookmakers and found anomalies in games where one team was a heavy favorite. Specifically, there were too few close wins by heavy favorites and too many wins where the favorite just failed to cover the spread.

  • This was similar to anomalies found by statisticians Adolphe Quetelet and Henri Poincare in other contexts and suggested the possibility of game fixing without endangering the outcome - heavy favorites could underperform just enough to fail to cover the spread.

  • While Wolfers’ work did not prove game fixing, it raised suspicions that in some small percentage of games, players may have been taking bribes to “shave points” or intentionally underperform.

  • Quetelet pioneered the use of statistics and the normal distribution curve to understand human behavior and society. He hypothesized the existence of an “average man” and sought to discover social “laws” governing how societies change over time and across cultures.

  • While Quetelet made important contributions by applying statistical thinking, his ideas of stable “social physics” were often unrealistic. Not all outcomes follow a normal distribution, and discovering laws was more difficult than anticipated. Nonetheless, his approach inspired later work in biology and physics.

  • Galton measured the life spans of sovereigns (kings/queens) and clergymen and found they were similar to other professions, leading him to conclude that prayer brought no extra benefits to longevity.

  • In his 1869 book, Galton put forth the idea that characteristics like height, head size, brain weight, etc. follow a normal distribution in populations and are determined by heredity.

  • He believed human character is also determined by heredity and follows a normal distribution. Only about 250 out of every 1 million men have exceptional ability and become eminent.

  • Galton founded the field of eugenics, focused on improving humans through selective breeding. His ideas influenced but did not endorse the Nazi policies.

  • Galton’s studies of inheritance led him to discover the mathematical concepts of regression toward the mean and the correlation coefficient, both central to modern statistics.

  • Karl Pearson later developed the chi-square test based on Galton’s work, allowing quantification of how well data fits an expected distribution, and validating the use of statistics to test hypotheses.

  • Through Galton, Adolphe Quételet’s work on data distributions and averages influenced the development of statistical physics by scientists like Maxwell and Boltzmann.

  • In 1848, two teenage girls in Hydesville, New York reported hearing unexplained knocking noises in their home, which had a reputation for being haunted. Through a code of knocks, they claimed to communicate with the spirit of a murdered peddler. This launch modern spiritualism and the belief that the dead can communicate with the living.

  • By the early 1850s, practices like table rapping, table moving, and table turning had become popular in the US and Europe. Groups would gather around a table and wait for it to move or rap on its own.

  • In 1853, scientists began investigating table moving due to its popularity. Some physicians noted an “unconscious consensus” seemed to form about how the table would move during the silent waiting period. When they disrupted this consensus, the table did not move.

  • Another trial with sitters expecting different directions of movement also failed to produce table movement. The physicians concluded the motions were due to unconscious muscular action by the sitters.

  • The definitive investigation was done by physicist Michael Faraday, suggesting the summary focused on his findings next. However, no further details about Faraday’s investigation are provided in this excerpt.

Based on the passage, here is a one paragraph summary:

Michael Faraday was one of the founders of electromagnetic theory and invented the electric motor. He conducted experiments on table turning, a phenomenon where tables would move apparently on their own during séances. Through ingenious experiments using an indicator to alert subjects when their hands were moving the table unconsciously, Faraday discovered that the subjects were unconsciously pushing and pulling the table through small movements, and their perception of patterns in the random movement created a self-fulfilling prophecy that the table was moving under its own power. His work showed that human perception is an act of imagination that fills in gaps, and reality is partly constructed in the mind of the observer rather than a direct representation of objective events.

  • Mathematician George Spencer-Brown noted that in a truly random sequence of 101,000,007 zeros and ones, you would expect to see at least 10 stretches of 1 million consecutive zeros. This illustrates that random patterns can appear non-random to the untrained eye.

  • Apple initially employed a random song shuffling method for iPods that sometimes led to repetitiveness, violating users’ expectations of randomness. The company then adjusted the algorithm to feel less random while actually being more random.

  • Studies have found that financial analysts’ recommendations and stock-picking abilities do not consistently outperform the market and are often just a result of chance. However, people still pay fees believing they can gain an edge.

  • Columnist Leonard Koppett accurately predicted the stock market for 18 of 19 years based on Super Bowl results, but this was purely due to chance given his methodology was just picking the winning NFL or AFL team.

  • Mutual fund manager Bill Miller had a 15-year streak of outperforming the S&P 500 index, leading many to believe in his “hot hand.” However, academic research shows success streaks in sports and other random processes are also often just a result of chance.

In summary, the passage discusses how randomness can appear patterned to the human eye and how chance success streaks are often misattributed to genuine skill or ability, especially in domains like sports, finance and prediction markets. Multiple studies are cited showing the limitations of our intuitions about randomness.

Here are the key points:

  • Bill Miller achieved an impressive streak of beating the stock market for 15 consecutive years. However, when looking at probabilities, such a streak could plausibly occur through random chance given the large number of fund managers over time.

  • Calculating the odds of Miller specifically beating the market for 15 years is very low (372,529 to 1). But considering thousands of managers over decades, the odds of someone achieving such a streak are much higher (around 75%).

  • Random coin tosses or other random processes can produce streaks and patterns that seem non-random. With a large enough sample size over long periods, remarkable streaks are likely to occur for someone purely by chance.

  • We tend to see patterns even in random data and assign meaning to them. Examples discussed include bombing clusters in London during WWII and cancer clusters - both could plausibly arise from random distributions.

  • Miller’s streak, while impressive, was not so improbable that it required skill to achieve given the number of managers and time periods involved. Random chance alone could produce such results.

  • Cancer registries that track rates of different cancers in geographical areas will often find statistically significant elevations of cancer in some areas due purely to random chance. Looking at a large number of small areas increases the likelihood of finding apparent clusters.

  • Drawing boundaries around an area after cancers are identified (“sharpshooting”) makes clusters seem more meaningful than they are. Increased availability of data online has led to more reports of cancer clusters being identified this way.

  • For most identified clusters to truly be due to environmental causes, exposure levels would need to be extremely high, comparable to chemotherapy. Nevertheless, people resist the explanation that clusters are random, leading to many investigations that find no underlying environmental cause.

  • Our desire for control makes it difficult to recognize randomness. Giving people even an illusion of control, like choosing lottery cards, influences their behavior even if it has no impact on outcomes. This helps explain why we resist seeing cancer clusters as random fluctuations.

  • The passage discusses the illusion of control and the confirmation bias - how people seek evidence to confirm their preexisting beliefs rather than challenging them. It gives examples of how this plays out in various situations.

  • One study showed that people rated academic studies supporting their own view on the death penalty more highly, even when all studies had the same methodologies. Reading the studies actually polarized and strengthened peoples’ existing beliefs.

  • The confirmation bias has negative consequences, like teachers focusing on evidence that confirms their initial views of students, and interviews where people look for reasons to confirm a first impression.

  • We are good at pattern recognition but focus more on confirming patterns than minimizing false conclusions. Chance events also produce patterns that we can misinterpret as meaningful.

  • Overcoming biases requires realizing chance produces patterns, questioning our perceptions, and spending equal time looking for evidence we are wrong as looking for reasons we are right.

  • The chapter transitions to discussing determinism and free will versus randomness in human destiny and achievement. It questions how predictable the future really is given chance influences and our limited knowledge of complex systems like human behavior and societies.

  • The passage describes how Edward Lorenz discovered the butterfly effect while running a weather simulation on a computer. He started the simulation midway using initial conditions from a previous printout, but the results diverged wildly due to small differences in the data.

  • This showed that tiny differences in initial conditions can lead to dramatically different outcomes over time (analogous to a butterfly flapping its wings causing large weather changes later). Lorenz’s discovery of this phenomenon was itself an example of the butterfly effect.

  • The passage then discusses how human affairs are unpredictable due to their complex, irrational nature as well as our inability to precisely know all initial conditions like laws of nature. This makes determinism an inadequate model for human experiences and futures.

  • The rest of the passage uses examples like a molecule moving in water and games of chess to illustrate how the past can seem obvious in hindsight even when the future was unpredictable. It was difficult to foresee events like Pearl Harbor based on the available information at the time, though explanations emerged later. This fundamental asymmetry between predicting the future vs. understanding the past is a common phenomenon.

  • The passage discusses how looking back at past mutual fund performance data, clear patterns seem to emerge, but these patterns have little predictive power for future performance.

  • Two graphs are shown - one plotting fund performance from 1991-1995 and ranking them, and another showing how those same funds ranked based on their prior performance did from 1996-2000. The order is largely randomized, showing past success is a poor indicator of future outcomes.

  • People underestimate the role of chance and randomness in outcomes. Explanations constructed after the fact give an illusion of predictability but have little relevance for forecasting.

  • Similar issues arise in other domains like business planning, where unforeseen changes undermine precise projections. Historians also warn against seeing the past as inevitable.

  • Hindsight bias leads us to believe we understand why past events unfolded as they did, but forecasting remains very difficult due to randomness and alternative possibilities prior to outcomes being realized. The crystal ball view is only possible looking back, not ahead.

So in summary, the passage cautions against overinterpreting patterns in past performance data and stresses the limits of predictability due to randomness, challenging the illusion that the past reveals how the future will unfold.

  • A valve at a nuclear power plant was accidentally left closed after maintenance, causing pumps to uselessly pump water toward a dead end.

  • Additionally, a pressure relief valve and gauge failed to detect the closed valve issue.

  • Individually these failures were common and acceptable, but together they led to a serious accident at Three Mile Island nuclear power plant.

  • This string of small failures led sociologist Charles Perrow to develop “normal accident theory,” which posits that in complex systems, seemingly minor issues can by chance combine to cause major incidents that are difficult to foresee or attribute to clear causes.

  • Similarly, economists argue that in markets, small random factors can accumulate over time through positive feedback loops to determine which companies come to dominate, rather than it just being down to intrinsic qualities.

  • Research on music downloads found popularity varied widely between different “worlds” and was influenced more by early random downloads than song quality alone.

  • The story of actor Bruce Willis landing the role of Moonlighting through a chance trip to LA illustrates how major life events and successes can stem from small random factors and unintended consequences.

  • The article discusses how Bill Gates went from a small software entrepreneur to becoming the richest man in the world through his founding of Microsoft.

  • It recounts how Gates came to license the operating system that became DOS to IBM after the company failed to reach an agreement with another programmer. This allowed Microsoft to dominate the PC market as more software developers wrote for DOS.

  • However, the article questions whether Gates would have become so wealthy and powerful if not for some random factors like the other programmer refusing to sign an agreement with IBM. His success may have been influenced by luck as much as his own skills and vision.

  • More broadly, it examines how society often wrongly attributes wealth and success purely to individual talent and hard work, rather than acknowledging the role of chance. People also tend to perceive the wealthy as more deserving and talented than others based solely on their outcomes.

  • Psychological experiments discussed showed that subjects rated individuals who received higher random pay as having performed better and having more worthy ideas, even when their actual performance was the same. This illustrates our bias towards inferring skills from wealth.

  • An experiment by psychologist David Lerner found that observers tended to view victims more negatively the more they suffered, assuming the victim was at fault. When told the victim would be compensated, observers did not develop negative views of the victim.

  • We tend to see causality and attribute success to talent or failure to lack of talent,reinforcing our perceptions. In reality,there is often little difference in ability between hugely successful and less successful people.

  • An experiment by Rosenhan found that even sane people were misdiagnosed as mentally ill when admitted to psychiatric hospitals based on staff perceiving everything through the lens of mental illness.

  • Expectations shape our perceptions in many areas like judging job applicants, car mechanics repairing visible rather than internal issues, and teachers grading identical work differently based on the student’s perceived ability.

  • Marketers exploit this by establishing brand expectations, as seen in studies finding little real difference in taste between cheap and premium vodka brands when tasted blindly.

  • Similarly, submissions of acclaimed novels under unknown author names were overwhelmingly rejected, showing success shapes perceptions of quality far more than the work itself.

  • The physicist mentioned did not succeed in his career and as a result many people saw him as a “crackpot”. However, the author believes he and his colleague John were brilliant physicists who courageously worked on theories that were out of favor, without any promise of imminent success.

  • The author argues that scientists and others striving to achieve should be judged more by their abilities than by their success alone. The connection between ability and success is loose and influenced by chance.

  • It is easy to admire the most successful people and look down on the unsuccessful, but success does not guarantee ability and vice versa. Chance plays a large role in success.

  • The author’s mother told a story of her sister Sabina who died in a concentration camp, despite making plans to stay safe. This taught the author that while planning is important, we can’t control the future or predict random events that may affect us. We should appreciate our good fortune and accept unexpected events, both positive and negative.

So in summary, the passage discusses the role of chance in success and abilities, argues against solely judging people by their results, and emphasizes appreciating good luck and accepting random life events.

Here is a summary of the source “y,” Journal of Political Economy 58, no. 3 (June 1950): 213:

This source is a journal article published in the Journal of Political Economy in June 1950 on page 213. As the title is not provided, no further specifics can be gleaned about the content or argument of the article from the bibliographic reference alone. The Journal of Political Economy is a respected academic journal that publishes research on various topics related to economics and politics. However, without more context about the article referenced here simply as “y,” it is difficult to infer much about its main ideas or conclusions. The reference provides basic bibliographic details like the author, title, journal, volume, issue, page number and date to uniquely identify the source, but not enough information to summarize the article’s focus or findings.

Here are summaries of the sources:

  1. Interview with Darrell Dorrell on August 1, 2005. No other details are provided.

  2. Wall Street Journal article from July 10, 1995 profiling a scholar who uses math to detect financial fraud.

  3. Passage from Writings of Charles S. Peirce in 1982, page 427. No other context is given.

  4. Reference to the Rand Corporation’s 1955 publication A Million Random Digits with 100,000 Normal Deviates and a 1982 journal article discussing induction and randomness.

  5. Account of Joseph Jagger’s roulette wheel predictions from a 1997 newspaper article.

  6. Details on the Bernoulli family, especially Jakob Bernoulli, drawing from several academic publications from 1949, 1978, and 1996.

  7. Quote about Jakob Bernoulli drawn from a 1978 book.

  8. Discussion of Jakob Bernoulli’s contributions drawing from 1986 and 1978 books.

  9. Brief biographical details about scientist Johann Bernoulli from a 2001 book.

  10. Quote from Johann Bernoulli drawn from a 1978 book.

  11. Discussion of statistics textbook from 1998.

  12. Quotes from 1975 and unknown date works discussing the emergence of probability.

  13. 1971 journal article on the law of small numbers.

  14. Quote from 1974 book discussing Jakob Bernoulli.

Here are summaries of the sources:

  1. This source analyzed how novice and expert wine tasters describe wines differently. Experts use more detailed descriptive terms while novices rely more on general impressions.

  2. This paper argues that expert opinion on wine value or quality is generally not very useful or predictive. Experts tend to rely more on subjective impressions rather than objective analysis.

The other sources listed did not have summaries provided and so were not included. Let me know if you would like me to expand on any of the summarized sources.

Here are summaries of three sources cited in the passage:

  1. “Performance of Investment Newsletters” (discussion paper no. 1805, Harvard Institute of Economic Research, Cambridge, Mass., November 1997). This paper analyzes the performance of investment newsletters over time. It likely finds that their performance is no better than random chance, calling into question their ability to consistently beat the market through their stock picks.

  2. James J. Choi, David Laibson, and Brigitte Madrian, “Why Does the Law of One Price Fail? An Experiment on Index Mutual Funds” (working paper no. W12261, National Bureau of Economic Research, Cambridge, Mass., May 4, 2006). This working paper presents an experiment on index mutual funds that aims to understand why the “law of one price” fails in financial markets. The law of one price states that identical goods should have only one single price. The experiment may show how behavioral biases can cause deviations from this theory.

  3. Leonard Koppett, “Carrying Statistics to Extremes,” Sporting News, February 11, 1978. This newspaper article discusses a statistics-based system for assessing basketball players created by sports journalist Leonard Koppett. It notes there was debate over whether the system truly failed or succeeded based on different definitions.

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