Read the book “The Drunkard’s Walk – How Randomness Rules Our Lives” by Mlodinow and pay special attend to the following questions. Some of these questions may appear on quizzes and exams.
Chapter 1 Peering through the Eyepiece of Randomness
1. Explain the phenomenon “regression toward the mean.”
In any series of random events an extraordinary event is most likely to be followed, due purely to chance, by a more ordinary one.
2. What factors determine whether a person will be successful in career, investment, etc.? Success in our careers, in our investments, and in our life decisions, both major and minor—is as much the result of random factors as the result of skill, preparedness, and hard work.
3. Was Paramount’s firing of Lansing the correct decision? After she was fired, Paramount films market share rebounded. No, Lansing was fired because of industry’s misunderstanding of randomness and not because of her own flawed decision making. Lansing had good luck at the beginning and bad luck at the end.
Chapter 2 The Laws of Truths and Half-Truths
1. What coined the term probability, or probabilis? (Latin: probabilis credible) Cicero’s principal legacy in the field of randomness is the term he used, probabilis, which is the origin of the term we employ today. But it is one part of the Roman code of law, the Digest, compiled by Emperor Justinian in the sixth century, that is the first document in which probability appears as an everyday term of art
2. What is the rule for compounding probabilities? How to compute probability that one event and another event both happening? According to the correct manner of compounding probabilities, not only do two half proofs yield less than a whole certainty, but no finite number of partial proofs will ever add up to a certainty because to compound probabilities, you don’t add them; you multiply. That brings us to our next law, the rule for compounding probabilities: If two possible events, A and B, are independent, then the probability that both A and B will occur is equal to the product of their individual probabilities.
3. Is the Roman rule of half proofs: two half proofs constitute a whole proof, correct? What do two half proofs constitute by the rule of compounding probabilities? 4. Suppose an airline has 1 seat left on a flight and 2 passengers have yet to show up. If there is a 2 in 3 chance a passenger who books a seat will arrive to claim it, what is the probability that the airline will have to deal with an unhappy customer? What is the probability that neither customer will show up? What is the assumption?
What is the probability that either both passengers or neither passenger will show up? 5. In DNA testing for legal trial, there is 1 in 1 billion accidental match and 1 in 100 laberror match. What is the probability that there is both an accidental match and a lab error? What is the probability that one error or the other occurred? Which probability is more relevant?
Chapter 3 Finding Your Way through a Space of Possibilities
1. What is “sample space”?
2. What is Cardano’s law of the sample space? (P. 62)
3. In the Monty Hall problem, why should the player switch after the host’s intervention? Chapter 4 Tracking the Pathways to Success
1. The grand duke of Tuscany’s problem: what is the probability of obtaining 10 when you throw three dice? What about 9?
2. What is Cardano’s law of the sample space?
3. What is the application of Pascal’s triangle?
4. For the Yankees-Braves World Series example, for the remaining 5 games, what is the probability that the Yankees win 2 games? 1 game?
5. What is mathematical expectation?
6. Explain why a state lottery is equivalent to: Of all those who pay the dollar or two to enter, most will receive nothing, one person will receive a fortune, and one person will be put to death in a violent manner?
Chapter 5 The Dueling Laws of Large and Small Numbers?
1. What is Benford’s law? Discuss some applications in business. 2. Explain the difference between the frequency interpretation and the subjective interpretation of randomness.
3. Do psychics exist?
4. What is tolerance of error, tolerance of uncertainty, statistical significance? 5. Describe some applications from the book of the law of large numbers and the law of small numbers.
Chapter 6 Bayes’s Theory
1. Two-daughter problem
In a family with two children, what are the chances that both children are girls? Ans: 25%
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? Ans 33%
In a family with two children, what are the chances, if one of the children is a girl named Florida, that both children are girls? Ans: 50%
2. How to apply Bayes’s Theory to determine car insurance rates? Ans : Models employed to determine car insurance rates include a mathematical function describing, per unit of driving time, your personal probability of having zero, one, or more accidents. Consider, for our purposes, a simplified model that places everyone in one of two categories: high risk, which includes drivers who average at least one accident each year, and low risk, which includes drivers who average less than one. If, when you apply for insurance, you have a driving record that stretches back twenty years without an accident or one that goes back twenty years with thirty-seven accidents, the insurance company can be pretty sure which category to place you in.
But if you are a new driver, should you be classified as low risk (a kid who obeys the speed limit and volunteers to be the designated driver) or high risk (a kid who races down Main Street swigging from a half-empty $2 bottle of Boone’s Farm apple wine)? Since the company has no data on you—no idea of the “position of the first ball”—it might assign you an equal prior probability of being in either group, or it might use what it knows about the general population of new drivers and start you off by guessing that the chances you are a high risk are, say, 1 in 3. In that case the company would model you as a hybrid—one-third high risk and two-thirds low risk—and charge you one-third the price it charges high-risk drivers plus two-thirds the price it charges low risk drivers. Then, after a year of observation—that is, after one of Bayes’s second balls has been thrown—the company can employ the new datum to reevaluate its model, adjust the one-third and two-third proportions it previously assigned, and recalculate what it ought to charge. If you have had no accidents, the proportion of low risk and low price it assigns you will increase; if you have had two accidents, it will decrease.
The precise size of the adjustment is given by Bayes’s theory. In the same manner the insurance company can periodically adjust its assessments in later years to reflect the fact that you were accident-free or that you twice had an accident while driving the wrong way down a one way street, holding a cell phone with your left hand and a doughnut with your right. That is why insurance companies can give out “good driver” discounts: the absence of accidents elevates the posterior probability that a driver belongs in a low-risk group.
3. Probability of correct diagnosis
Suppose in 1989, statistics from the Centers for Disease Control and Prevention show about 1 in 10,000 heterosexual non-IV-drug-abusing white male Americans who got tested were infected with HIV. Also suppose about 1 person out of every 10,000 will test positive due to the presence of the infection. Suppose 1 in 1,000 will test positive even if not infected with HIV (false positive). What is the probability that a patient who tested positive is in fact healthy?
Ans: So if you test 10 000 people you will have 11 positives – 1 who is really infected, 10 are false positives. Of the 11 positive testees, only 1 has HIV, that is, 1/11. Therefore the probability that a positive testee is healthy = 10 / 11 = 90.9%
4. O. J. Simpson trial
According to FBI statistics, 4 million women are battered annually by husbands and boyfriends in U.S. and in 1992 1,432 or 1 in 2500 were killed by their husbands or boyfriends. The probability that a man who batters his wife will go on to kill her is 1 in 2500. The probability that a battered wife who was murdered was murdered by her abuser is 90%. Which probability is relevant to the O. J. trial? What is the fundamental difference between probability and statistics?
Ans: 1) Relevant one is the probability that a battered wife who was murdered was murdered by her abuser = 90%. 2)the fundamental difference between probability and statistics: the former concerns predictions based on fixed probabilities; the latter concerns the inference of those probabilities based on observed data.
Chapter 7 Measurement and the Law of Errors
Why did the author argue that “when elections come out extremely close, perhaps we ought to accept them as is, or flip a coin, rather than conducting recount after recount?” Ans: (pg= 127 and 128) Elections, like all measurements, are imprecise, and so are the recounts, so when elections come out extremely close, perhaps we ought to accept them as is, or flip a coin, rather than conducting recount after recount.
2. What is mathematical statistics?
Ans: Mathematical statistics, provides a set of tools for the interpretation of the data that arise from observation and experimentation. Statisticians sometimes view the growth of modern science as revolving around that development, the creation of a theory of measurement. But statistics also provides tools to address real-world issues, such as the effectiveness of drugs or the popularity of politicians, so a proper understanding of statistical reasoning is as useful in everyday life as it is in science.
3. Wine tasting
Should we believe in wine ratings from those “wine experts”? Why or why not?
Two groups wine tasting experts produce the following results: (a) 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
(b) 80 81 82 87 89 89 90 90 90 91 91 94 97 99 100
Compare the two groups of data. (pg 134)
From the theoretical viewpoint, there are many reasons to question the significance of wine ratings. For one thing, taste perception depends on a complex interaction between taste and olfactory stimulation. Strictly speaking, the sense of taste comes from five types of receptor cells on the tongue: salty, sweet, sour, bitter, and umami. The last responds to certain amino acid compounds (prevalent, for example, in soy sauce). But if that were all there was to taste perception, you could mimic everything—your favorite steak, baked potato, and apple pie feast or a nice spaghetti Bolognese—employing only table salt, sugar, vinegar, quinine, and monosodium glutamate.
Fortunately there is more to gluttony than that, and that is where the sense of smell comes in. The sense of smell explains why, if you take two identical solutions of sugar water and add to one a (sugar-free) essence of strawberry, it will taste sweeter than the other.15 The perceived taste of wine arises from the effects of a stew of between 600 and 800 volatile organic compounds on both the tongue and the nose.16 That’s a problem, given that studies have shown that even flavor-trained professionals can rarely reliably identify more than three or four components in a mixture
4. Can professional mutual fund managers (stock pickers) beat students who pick stocks by tossing coins?
5. What is the margin of error in a poll? Should variation within the margin of error be ignored in a poll?
Ans: < 5% (or 3.5%). Yes, any variation within the margin of error should be ignored in a poll
6. What is the central limit theorem?
Ans: The probability that the sum of a large number of independent random factors will take on any given value is distributed according to the normal
Chapter 8 The Order in Chaos
1. Who are the founders of statistics?
Graunt and his friend William Petty have been called the founders of statistics, a field sometimes considered lowbrow by those in pure mathematics owing to its focus on mundane practical issues, and in that sense John Graunt in particular makes a fitting founder.
2. How did Graunt estimate the population of London in 1662? What is Graunt’s legacy? From the bills of mortality, Graunt knew the number of births. Since he had a rough idea of the fertility rate, he could infer how many women were of childbearing age. That datum allowed him to guess the total number of families and, using his own observations of the mean size of a London family, thereby estimate the city’s population. He came up with 384,000— previously it was believed to be 2 million.
Graunt’s legacy was to demonstrate that inferences about a population as a whole could be made by carefully examining a limited sample of data. But though Graunt and others made valiant efforts to learn from the data through the application of simple logic, most of the data’s secrets awaited the development of the tools created by Gauss, Laplace, and others in the nineteenth and early twentieth centuries.
3. How did Poincare show the baker was shortchanging customers? French mathematician Jules-Henri Poincaré employed Quételet’s method to nab a baker who was shortchanging his customers. At first, Poincaré, who made a habit of picking up a loaf of bread each day, noticed after weighing his loaves that they averaged about 950 grams instead of the 1,000 grams advertised. He complained to the authorities and afterward received bigger loaves.
Still he had a hunch that something about his bread wasn’t kosher. And so with the patience only a famous—or at least tenured—scholar can afford, he carefully weighed his bread every day for the next year. Though his bread now averaged closer to 1,000 grams, if the baker had been honestly handing him random loaves, the number of loaves heavier and lighter than the mean should have diminished following the bellshaped pattern of the error law. Instead, Poincaré found that there were too few light loaves and a surplus of heavy ones. He concluded that the baker had not ceased baking underweight loaves but instead was seeking to placate him by always giving him the largest loaf he had on hand.
4. Are all data in society such as financial realm normal? (Yes) Are film revenue data normal? (No) For one thing, not all that happens in society, especially in the financial realm, is governed by the normal distribution. For example, if film revenue were normally distributed, most films would earn near some average amount, and two-thirds of all film revenue would fall within a standard deviation of that number.
But in the film business, 20 percent of the movies bring in 80 percent of the revenue. Such hit-driven businesses, though thoroughly unpredictable, follow a far different distribution, one for which the concepts of mean and standard deviation have no meaning because there is no “typical” performance, and megahit outliers, which in an ordinary business might occur only once every few centuries, happen every few years.
5. Who dubbed the phenomenon “regression toward the mean”? Explain its meaning. Galton dubbed the phenomenon—that in linked measurements, if one measured quantity is far from its mean, the other will be closer to its mean—regression toward the mean.
6. Who coined the term “the coefficient of correlation”? Explain its meaning. Galton coined the term “the coefficient of correlation “.The coefficient of correlation is a number between −1 and 1; if it is near ±1, it indicates that two variables are linearly related; a coefficient of 0 means there is no relation.
7. Discuss the applications of the chi-square test?(Pg 165 166 167) Pearson invented a method, called the chi-square test, by which you can determine whether a set of data actually conforms to the distribution you believe it conforms to.
8. What is statistical physics?
James Clerk Maxwell and Ludwig Boltzmann, two of the founders of statistical physics. Statistical physics was aimed at explaining a phenomenon called Brownian motion. Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems.
9. What is a drunkard’s walk or random walk?
The random motion of molecules in a fluid can be viewed, as a metaphor for our own paths through life, and so it is worthwhile to take a little time to give Einstein’s work a closer look. According to the atomic picture, the fundamental motion of water molecules is chaotic. The molecules fly first this way, then that, moving in a straight line only until deflected by an encounter with one of their sisters. As mentioned in the Prologue, this type of path—in which at various points the direction changes randomly—is often called a drunkard’s walk, for reasons obvious to anyone who has ever enjoyed a few too many martinis (more sober mathematicians and scientists sometimes call it a random walk).
Chapter 9 Illusions of Patterns and Patterns of Illusion
1. What caused the table to move, spirit?
not a direct consequence of reality but rather an act of imagination.
2. What is significance testing?
Significance testing, was developed in the 1920s by R. A. Fisher, one of the greatest statistician for scientific research. It is a formal procedure for calculating the probability of our having observed what we observed if the hypothesis we are testing is true. If the probability is low, we reject the hypothesis. If it is high, we accept it.
3. Why did Apple founder Steve Jobs made the ipod’s shuffling feature “less random to make it feel more random”?
Spencer-Brown’s point was that there is a difference between a process being random and the product of that process appearing to be random. Apple ran
into that issue with the random shuffling method it initially employed in its iPod music players: true randomness sometimes produces repetition, but when users heard the same song or songs by the same artist played back-to-back, they believed the shuffling wasn’t random. And so the company made the feature “less random to make it feel more random,” said Apple founder Steve Jobs.
4. Suppose there are 1000 mutual fund managers picking stock for 15 consecutive years by each tossing a coin once a year. If a head is obtained, he/she beats the market (a fund manager either beats the market average or not). What is the probability that someone among the 1000 who would toss a head in each of the 15 years? From Nobel Prize-winning economist Merton Miller: “If there are 10,000 people looking at the stocks and trying to pick winners, one in 10,000 is going score, by chance alone, and that’s all that’s going on.
It’s a game, it’s a chance operation, and people think they are doing something purposeful but they’re really not.” Ans: The chances that, after fifteen years, a particular coin tosser would have tossed all heads are then 1 in 32,768. But the chances that someone among the 1,000 who had started tossing coins in 1991 would have tossed all heads are much higher, about 3 percent.
5. What is confirmation bias?
When we are in the grasp of an illusion—or, for that matter, whenever we have a new idea—instead of searching for ways to prove our ideas wrong, we usually attempt to prove them correct. Psychologists call this the confirmation bias, and it presents a major impediment to our ability to break free from the misinterpretation of randomness.
Chapter 10 The Drunkard’s Walk
1. What is the butterfly effect?
The butterfly effect, based on the implication that atmospheric changes so small they could have been caused by a butterfly flapping its wings can have a large effect on subsequent global weather patterns. 2. Can past performance of mutual fund managers predict future performance? No.