Chapter 14 Chapter 10 The Achilles Heel of Human Cognition—Probabilistic Reasoning
Q: Men are taller than women, right?
answer."
Q: All men are taller than all women, right?
Answer: "Wrong."
Completely correct.Believe it or not, in this chapter we will also spend some time discussing some of the questions you already know, as you can see from the answers to the two questions above, you already know some of the answers, but first Don't skip this chapter just because of that.Because surprises await you in what follows as we explain some seemingly very simple principles.
You gave an affirmative answer to the first question because you did not understand the sentence "men are taller than women" as the second sentence says "all men are taller than all women".You correctly understood the first question to mean "men tend to be taller than women", because everyone knows that not all men are taller than all women.You understand that the question reflects a probability trend rather than a fact that applies in every situation.What we mean by a probability trend is that there is a greater probability, but not necessarily so, in all cases.That is to say, the relationship between gender and height should be described in terms of possibility and probability, not in terms of inevitability.The nature of many relationships in nature is also probabilistic, for example: areas close to the equator are hotter; the number of children per family does not exceed 8; insects outnumber humans in most parts of the earth.These are statistically verifiable trends, but not every word of them is absolute, and there may still be exceptions.Because they are trends and laws of probability, not relationships that hold in all cases.
Virtually all facts and relationships revealed by psychological science are expressed in terms of probabilities.Nor is this unique to psychology.In other disciplines, many laws and relationships are also expressed in terms of probability rather than certainty.For example, all subdisciplines of population genetics are based on probability relationships; physicists tell us that the distribution of electron loads in atoms is also described by probability functions.
Indeed, most probabilistic trends revealed by psychology are weak.In psychology, behavioral relationships are described probabilistically, but this fact does not make it very different from other sciences.As Jacob Bronowski noted (1978a), many people still cannot accept the fact that as science continues to open up new fields of study, more and more of the laws of science are will be described in probabilistic form:
In this chapter, we want to make you as comfortable as possible in the "world of sometimes and maybe" because, if one wants to understand psychology, one must understand the "probabilistic reasoning" of this chapter. The theme is at ease.
Much of the public is aware that many conclusions in medicine are expressed in terms of probabilistic trends rather than absolute certainties.Smoking can cause lung cancer and trigger other health problems.The relevant medical evidence is overwhelming.But does every smoker get lung cancer?Do all quitters remove their lung cancer risk?Most people wouldn't think these inferences would hold up.Smoking greatly increases the probability of developing lung cancer, but not absolutely.Medicine can tell us with a good degree of confidence that people in a smoking group are more likely to die of lung cancer than people in a similar non-smoking group, but not which people will die. This relationship is probability; it Not applicable in all cases.We all know this - really?We often see a non-smoker trying to convince an addict to quit smoking, citing statistics that smoking causes lung cancer, only to get backlash: "Hey, stay away! Look at that Old Joe in the shop, he's been smoking three packs of Camels a day since he was 16! Now he's 81 and looking strong!" The inferences one might make about this are obvious: it's this one exception that's overturned relationship between smoking and lung cancer.
Surprisingly and frustratingly, this rebuttal has been tried and tested.It is often the case that whenever a single case is used to invalidate a trend in probability, many people tend to nod in agreement, reflecting a misapprehension of the nature of statistical laws.If people think that a special case can invalidate a law, they must think that the law should apply in all cases.
In short, they misunderstood the nature of the laws of probability.Even the strongest trends have a few "special cases" that go against it.Taking smoking as an example, only 5% of people who live to the age of 85 are smokers (University of California, Berkeley, 1991).Or to look at it another way, 95% of people who live to the age of 85 are never smokers, or smokers for a period of time but eventually quit.Continuous, uninterrupted smoking significantly shortens lifespan (Uaiversity of California, Berkeley, 1991), however, a minority of smokers live to the age of 85.
Borrowing the term from psychologists Nisbett & Ross (1980), we call stories like "Old Joe" the use of "so-and-so" statistics: since someone knows a "so-and-so "Go against a well-established statistical trend, and the trend is cast into doubt.For example, we often hear things like - "Are you saying that jobs in services are expanding while jobs in heavy industry are shrinking? That's not true, I just know that 'so and so' got a job at a steel mill last Thursday." Work"; "You said there are fewer children in the family than 30 years ago? Cut the bullshit! The young couple next door already has 3 children, but they are not yet 30 years old"; "You say that children usually tend to Belief in the religion of their parents? But as far as I know, the child of one of my colleagues converted to another religion just the other day."
When we are confronted with strong evidence that contradicts previously held beliefs, the ubiquitous "so-and-so" always jumps out at once to deny these statistical laws.Therefore, we can say that people actually know a lot, and they just use "so-and-so" as a tool to veto facts that are contrary to their ideas.However, findings from psychologists who study human decision-making and reasoning suggest that people use "so-and-so" not just because it's a useful debating device.On the contrary, the main reason why this false argument model is used so often is that people do not know how to deal with probability information.New research in the psychology of decision-making finds that probabilistic reasoning may be the Achilles' heel of human cognition.
Probabilistic thinking is involved in many fields, including science, technology, and personnel.So there is no particular reason to think that this kind of thinking is more important for understanding psychology than other disciplines.However, the results of psychological research are often misinterpreted because people have problems using probabilistic information.We all understand that "men are taller than women" is a statement of probability trends, so just because there is a special case (a certain man is shorter than a certain woman) does not mean that this statement is false.Many people can interpret the statement "smoking can cause lung cancer" in the same way, although "Old Joe" may still be persuasive to addicts who are unwilling to believe that smoking will kill them.Similar probabilistic statements about behavioral trends, however, have sparked widespread suspicion and are often discarded as soon as "so-and-so" shows up.Many psychology teachers tend to get the same reaction when they discuss the evidence for relationships between certain behaviors.For example, teachers can present the fact that children's academic performance correlates with the socioeconomic status of the family and the educational level of the parents.But this fact is often disputed by at least one student who will say that he has a friend who is a National Merit Scholar but whose father only graduated from high school.Even those who understand the smoking-lung cancer example have become vacillating on the issue.
People never thought of using "so-and-so" arguments to refute the findings of medicine and physics, but they used to refute the results of psychological research.Most people understand that the treatments, theories, and facts presented by medical science are probabilistic.For example, they understand that a drug does not work for a group of patients, and that medicine often cannot tell us in advance which patients a drug will work for.Generally speaking, if 100 patients receive a certain treatment plan and 100 patients do not receive any treatment, after a period of time, the condition of the 100 patients who receive treatment will be better than the condition of the 100 patients who do not receive treatment.No one doubts the value of this treatment because of this probabilistic statement that does not apply in all cases.A similar situation exists in many psychological research results and the effects of psychotherapy.However, when the results of psychological research and the effects of psychotherapy are not applicable in all situations, it often leads to great disappointment and contempt for psychology.When confronted with the topic of psychology, people often forget one of the most basic principles, that is, knowledge does not need to be completely certain to be useful—even if some knowledge cannot predict the specific situation of an individual, if it can predict the general trend of the group Having predictive power is also very beneficial.Predictions of outcomes based on population characteristics are often referred to as population statistics or statistical forecasts (the concept of statistical forecasts is discussed in detail in the next chapter).
People often set a higher standard for psychological prediction than other sciences.Think about it, when an unfit person goes to the doctor and the doctor says that unless he exercises and changes his diet, he is at high risk of having a heart attack.We don't think the doctor's information is useless just because the doctor didn't tell the person "he's going to have a heart attack on September 18, 2012, if he doesn't change his diet."We can easily understand that the doctor's prediction is probabilistic and cannot achieve that kind of precision.Likewise, when geologists tell us that there is an 80 percent chance that an earthquake of magnitude 8.0 or greater will occur in an area in the next 30 years, we don't think that because they didn't say "July 5, 2012 there will be Earthquake happened here" to belittle its knowledge.
Psychology, however, is often set to a higher standard.When school psychologists recommend a training program for children with learning disabilities, they are clearly making probabilistic predictions—the training will make these children more likely to do well.The situation is similar when a clinical psychologist recommends a program for children with self-harming behaviors.Psychologists judge that if the treatment is carried out according to the plan, there will be a high probability of obtaining a good result.But unlike the heart attack and earthquake examples, psychologists are often confronted with questions like, "But when will my child be able to read at a certain grade level?" or "How long will he be in this treatment plan?" class questions.None of these questions can be answered, just as when earthquakes and heart attacks occur is also unanswerable, because predictions are made for all of these problems—heart attacks, children with learning disabilities, earthquakes, and children who harm themselves It's all probabilistic.
For these reasons, a comprehensive appreciation of probabilistic reasoning is essential to understanding psychology.It is intriguing and somewhat ironic that psychology is probably the biggest victim of people's inability to think statistically, yet psychologists are the ones who have studied the human capacity for probabilistic reasoning the most.
Over the past 20 years, Daniel Kahneman of Princeton (2002 Nobel Laureate; see MacCoun, 2002), Richard Nisbett of the University of Michigan, and the late The research of psychologists such as Amos Tversky (Amos Tversky) has revolutionized our understanding of human reasoning ability.In their research, they found that many people had no basic principles of probabilistic reasoning at all in their minds, and many more had some but incomplete ones.As scholars have often pointed out, it is not surprising that these basic principles are not fully developed in people's minds.As a branch of mathematics, statistics is a relatively recent development (Hacking, 1975).Games of chance existed for centuries before the laws of probability were discovered.This is yet another example: personal experience is not enough for people to gain a basic understanding of the world (see Chapter 7).Formal studies of the laws of probability have uncovered how games of chance work, but thousands of gamblers, and their personal experiences, are not enough to reveal the nature of games of chance.
The problem is that the more complex society is, the more people need probabilistic thinking.If an ordinary person wants to have a basic understanding of the society in which he lives, he should have at least the most basic ability of statistical thinking.
You may have the following questions: "Why do they increase my insurance premiums? Why does Zhang San's premiums are higher than Li Si's? Is the Social Security Administration crazy? Is there a shady scene in our state's lottery? Is the crime rate increasing? Decrease? Why do doctors order these tests? Why do Europeans have access to some very rare drugs, but Americans can't? Do women really earn less than men for the same work? International trade really reduces the income of Americans. employment opportunities, and lower their wages? Is education in Japan better than ours? Is health care in Canada really better and less expensive than in America?" These are good questions to ask, and it's all about our society Concrete and practical questions of how it works.To know the answer to each question, we must use statistical thinking.
Clearly, due to space limitations in this book, it is impossible to discuss statistical thinking comprehensively.However, we will briefly discuss some common pitfalls in probabilistic reasoning.The best way to learn probabilistic thinking skills is to discover what mistakes people make most often in statistical reasoning.In addition, awareness of some myths is critical to understanding the importance of psychological findings and theories.
In the field of psychology, it is a well-established finding that information about a specific event can often completely beat out more abstract probability information (the "freshness" problem discussed in Chapter 4).Examples of ignoring probabilistic information abound, and are not limited to laymen with little scientific knowledge.Casscells, Schoenberger, & Graboys (1978) conducted a study in four teaching hospitals of Harvard Medical School in which they gave 20 medical students , 20 attending physicians, and 20 office workers asked the following series of questions: "If 1 person in every 1,000 people is living with the HIV virus (HIV), suppose there is a test that can diagnose 100% of the people who really carry the virus people with the virus; finally, assume that this test has a 5% positive misdiagnosis rate. That is, this test will also falsely detect 5% of the people who do not carry HIV as HIV carriers. Suppose we If a random person takes this test and the result is positive, it indicates that the person is HIV positive. Assuming we don't know the person's medical history, what is the probability that he is really HIV positive?
The common answer is 95%, and the correct answer is about 2%.Physicians overestimate the probability that a positive test result indicates disease because they overestimate the probability of disease that a positive test result really represents because they place too much emphasis on individual case information and ignore base rate information on the other hand.A little logical reasoning will illustrate the importance of base ratios for probabilities. Only 1 in 1000 people is truly HIV positive.If another 999 people (who were not sick) were also tested, close to 50 of them (999 multiplied by 0.05) would be found to have the virus due to the 5% false positive rate of this test.That would bring the number of people who tested positive to 51.Because only one of these 51 people is truly HIV-positive, the probability of this person being diagnosed with the disease is actually only close to 2%.In short, the base rate is that the vast majority of people do not carry the virus (1 in 1,000 carriers).This fact, combined with the established false-reporting rate, provides confidence that, in absolute numbers, the majority of people who test positive do not carry the virus.
Although the physicians involved in the study by Cassells et al. quickly realized the correctness of the above probabilistic logic, their initial gut reaction was to ignore the base rate and place too much weight on the clinically tested evidence.In short, the fact is that doctors know what's right, but instinctively draw the wrong conclusions.Psychologists refer to such problems as cognitive illusions (see Kahneman & Frederick, 2002, 2005).In a cognitive illusion, even when people know the correct answer, they can draw wrong conclusions because of the way the question is asked.
All the examples we've mentioned here are cognitive illusions because they exploit a flaw in human reasoning: placing too much weight on the evidence provided by individual events and neglecting statistical information.For most people, individual case evidence (research results of laboratories) seems to be tangible and concrete, while probabilistic evidence seems to be intangible and uncertain.Of course, this understanding is wrong, because case evidence itself must be probabilistic.A clinical test will misdiagnose a disease with a certain probability.The situation above is an example of a combination of two probabilities that must be considered in order to make the right decision - the probability of making a correct or incorrect diagnosis based on the individual case evidence (i.e. 95% or 5%) and the a priori provided by past experience. Probability (also called basis ratio).There are ways to integrate these probabilities that are right, wrong, and often wrong—especially when the individual case evidence gives the illusion of concreteness (recall the freshness problem discussed in Chapter 4) —People tend to integrate information in the wrong way.The failure of this kind of probabilistic reasoning can greatly hinder the application of psychological knowledge, because psychological knowledge often expresses the relationship between behaviors in the form of probability.
Popular science writer Cole (KCCole, 1998) let us imagine the following two situations.One is to use the mortality rate of smoking to persuade people not to smoke. For example, the mortality rate of smoking is 0.000055, which is the most common way of persuading people.The second method is a little more graphic, asking the smoker to imagine that one pack out of every 18,250 is special—it's filled with explosives that kill the smoker when he opens it.We absolutely know which one works better - yet they express the same truth.
Please consider the following two questions posed by Tversky and Kahneman (Tversky & Kahneman, 1974):
1. There are two large and small hospitals in a small town.About 45 babies are born every day in a large hospital, and about 15 babies are born a day in a small hospital.As you know, about 50% of babies are boys, but the exact percentage varies from day to day, sometimes it's higher than 50%, sometimes it's lower.Each hospital recorded the number of days in a year when the proportion of babies born was more than 60 percent male.Which hospital do you think recorded the most days?
a. Big hospital
b. Small hospital
c.Basically the same
2. Suppose a container is full of balls, 2/3 of which are of one color and the remaining 1/3 are of another color.A person takes 5 balls out of it and finds that 4 are red and 1 is white.Another person takes 20 balls out of it and finds that 12 are red and 8 are white.Who would be more confident that 2/3 of the balls in the container are red and 1/3 of the balls are white than that 1/3 of the balls are red and 2/3 of the balls are white ?What probabilities would these two give?
For the first question, most people answered "basically the same" and the remaining half chose large hospitals and half chose small hospitals. But the correct answer was small hospitals, so nearly 75% of the subjects gave wrong answers .The wrong answer is due to people's failure to realize the importance of sample size in this question. When other factors are held constant, a larger sample will always provide a more accurate estimate of the true value of the population. That is, On any given day, the probability of a boy birth is closer to 50% in larger hospitals due to the larger sample size. Conversely, smaller samples always tend to be farther from the overall mean. Therefore, small hospitals There will be many more days that record a ratio of male babies that contradicts the overall average (60%, 40%, 80%, etc.).
In answering the second question, the majority felt that a sample of 5 balls provided more convincing evidence that the majority of the balls in the container were red.In fact, the probability is just the opposite.For the 5-ball sample, the probability that most of the urns are red is 8:1.In a sample of 20 balls, the odds are 16:1.Although in the sample of 5 balls, the proportion of red balls caught is higher (80%:60%), consider that another sample is 4 times the size, so the proportion of balls can be made more precisely. estimate.However, most subjects were confused by the higher proportion of red balls in the 5-ball sample, and did not fully consider the greater confidence in the 20-ball sample.
A fundamental principle of evidence assessment in different domains is to recognize the effect of sample size on the credibility of information, which is particularly important for understanding findings in the behavioral sciences.Whether we realize it or not, we hold some common perceptions about larger groups.We seldom realize how flimsy a foundation of facts our strongest beliefs are based on.Put together the observations of a few neighbors and colleagues, and a few anecdotes we've seen on the TV news, and we can't wait to have an opinion on "human nature" or "Americans."
Please answer the following two questions:
Question A: Imagine that you are tossing an ordinary coin (the probability of heads and tails of the coin is 50% each), and there have been 5 heads in a row.For the 6th time, do you think
____ Tails are more likely than heads
____ heads are more likely than tails
____ Heads are equally likely to come up as tails
Question B: When playing slot machines, the chance of winning money is 1 in 10.Julie won the first 3 times.The odds of her winning next time are ____ in ____
These two questions are designed to detect whether you are prone to the so-called gambler's fallacy - the tendency to make a connection between past events and future events, when in fact the two are independent.The two outcomes are independent, and the occurrence of one event does not affect the probability of the other event occurring.Most games of chance are of this nature.For example, the numbers on the wheel of fortune are not related to the previous numbers.Half the roulette numbers are red and the other half are black (we'll ignore the green zero and double zero for simplicity), so for any given spin, there's an equal chance (0.50) of being red.However after 5-6 consecutive red numbers many bettors switch to black as they think black is more likely now.This is the gambler's fallacy: the belief that previous outcomes affect the probability of the next outcome, even though they are independent events.In this case, the bettors were wrong in their beliefs.Roulette doesn't remember what happened before.Even if 15 red numbers appear in a row, the probability of the red number appearing in the next round is still 0.50.
In problem A, some people think that after 5 occurrences of heads, tails are more likely.Thinking so, they fall into the gambler's fallacy.The correct answer is that heads and tails are equally likely to come up on the 6th run.Likewise, any answer to question B that is not 1/10 falls into the gambler's fallacy.
The gambler's fallacy is not limited to inexperienced gamblers.Research has shown that even experienced gamblers who gamble 20 hours a week still exhibit the gambler's fallacy (Petry, 2005; Wagenaar, 1988).In fact, studies have shown that individuals undergoing gambling addiction treatment believe in the gambler's fallacy more than controls (Toplak et al., inpress).
It is important for us to realize that this fallacy is not limited to games of chance, it exists anywhere where probability plays a significant role.In other words, it's in almost everything.An example is the genetic makeup of babies.Psychologists, doctors and marriage counselors often encounter couples who already have two girls and are planning to have a third child because "we want a boy, and this time it has to be a boy."This is the gambler's fallacy, the probability of having a boy after two girls is exactly the same (nearly 50%) as it was with the first child.Having two girls does not increase the probability that the third child will be a boy.
The gambler's fallacy exists anywhere there is an element of chance, such as sports games and the stock market.Several psychologists (Gilovich, Vallone, & Tversky, 1985; Burns, 2004) have studied the superstition in basketball of "shooting streaks" or "hot hands," which refers to the belief that a particular pitcher can Become "hot" and after making consecutive shots, the next shot will be more likely ("pass the ball to him, he's hot now").Researchers have confirmed that basketball players and fans alike strongly believe in "continuous hits."For example, in one survey, 91% of basketball fans believed that a player who had just made two or three shots had a higher chance of making his next shot than a player who had just made two or three turnovers ; 84% of fans believe that it is important to pass the ball to the player who has just made two or three consecutive shots.Ask fans to estimate, assuming a player has a 50% shooting percentage on the field, what is the probability of the next shot after he makes or misses a shot.As a result, fans estimated the former at 61% and the latter at 42%.Researchers surveying players on the Philadelphia 76ers basketball team found that most (but not all) players held belief in streaks almost as strongly as fans did (see Gilovich et al., 1985).
But why are we discussing streaks under the heading of the gambler's fallacy?Because there is no such thing as consecutive shots!Gilovich et al. (1985) studied field goal statistics for the Philadelphia 76ers and Boston Celtics during the 1980-1981 season.Players' shots haven't been contextualized this season.Let's see what this means from a non-technical perspective.
The gambler's fallacy believes that independent events are related, that is, there is a dependency relationship between unrelated events.From a statistical point of view, continuous shooting can be interpreted as that after two or three consecutive shots, the shooting percentage will be higher than that of the previous few missed shots.Gilovich et al. (1985) calculated this probability and found no evidence to support this hypothesis.For example, the stats for Julius Irwin (the player with the most field goals on the Philadelphia 76ers) show that after making three consecutive field goals, he has a 0.48 percentage on subsequent field goals, while three consecutive missed shots After two consecutive hits, the next hit rate is 0.52, and after two consecutive misses, the next hit rate is 0.51; after one shot, the next hit rate is 0.53, after one miss After the middle, the next hit rate is 0.51.Simply put, Irwin's shooting percentage is close to .50 regardless of his first few shots -- there's no such thing as a streak.
The profiles of other players are also very similar.Lionel Hollins was 0.46 on the next shot after two consecutive hits and 0.49 on the next after two consecutive misses.After he makes one, his subsequent field goal percentage is 0.46, exactly the same as after a miss.What this means is that regardless of the results of Hollins' first few shots, he's always shooting closer to 47 percent.The Boston Celtics free-throw profile says the same thing.For example, Larry Bird makes his next free throw 88 percent of the time after making a free throw, and 91 percent of his next free throw after missing one.Nat Archibald made his next free throw 83 percent of the time after making a free throw, and 82 percent of his next free throw after a miss.It can be seen that there is no continuous shooting in free throws.The belief that players can get "hot hands" is indeed an example of the gambler's fallacy, that is, the belief that there is a connection between in fact independent, unrelated events.
Interestingly, the gambler's fallacy appears to be an example of the "intuitive physics" discussed in Chapter 6—the inability of experience alone to tell people the truth about the world.Gilovich et al. (1985) tested the performance of college basketball players practicing 15-yard field goal attempts on an empty court (ie, without any defenders).They made these players bet on 100 field goal percentages.The players should definitely win because they generally hit about 50% of the time at this distance, and the betting rule is that the player wins a little more when he hits than loses when he misses.However, players can choose to bet more (so they win more and lose more) or bet less (so they win less and lose less) before each shot.Obviously, players will win more if they can predict their own performance.That is to say, when they think the probability of hitting is high, they will choose to bet more; and when they think the probability of hitting is low, they will choose to bet less.Experimental results show that even professional players do not notice the phenomenon of "hot hands": after making one or more shots, the probability of making another shot is no higher than that of making another shot after missing one.However, the players all believed that there was a situation similar to "hot hands".After making a shot, they bet more on the next shot than after a miss.It turned out that the players couldn't predict their own performance at all: they predicted outcomes no better than random.
The gambler's fallacy stems from many misunderstandings about probability.One such misconception is that if a process is truly random, there can be no sequence that repeats the same outcome or pattern, even for a trivial random event (e.g., six coin flips).People habitually underestimate the possibility of repetitions (forwards and positives) or patterns (forwards and wrongs and positives and negatives) occurring in a random sequence.Because of this, when people simulate a set of truly random sequences, it often backfires to produce a permutation with few repetitions and certain patterns.This is because people often mistakenly let the possible results appear in turn as much as possible, thinking that this is called random sampling, which undoubtedly destroys the structure that may appear in a truly random permutation (Nickerson, 2002; Towse & Neil, 1998 ).
People who claim to have psychic abilities can easily take advantage of this delusion.Such a demonstration is often carried out in college psychology classes. The teacher asks a student to prepare an arrangement of 200 numbers, and the 200 numbers are randomly and repeatedly drawn from the three numbers 1, 2, and 3.After you finish, don't let the teacher see it.Next, have the student concentrate on the first number he writes while the teacher guesses what the number is.After the teacher states his guess, the student announces the correct answer to the class and the teacher.Someone records the number of correct guesses until the 200 numbers are guessed.Before the experiment started, the teacher claimed to have psychic ability and could use mind reading to prove the existence of "psychic ability" during the experiment.Usually, before showing it, the teacher would ask the class how much of his guess—that is, what percentage of his guesses were “right”—to prove that he did have psychic abilities.At this time, a student who has taken a statistics course usually responds that since purely random guessing can also guess 33% of the time, it must be more than 33% correct in order to convince others that he has psychics. , reaching at least 40%.Most of the students in the class would agree with this point of view.After the demonstration, it turned out that the teacher's guess rate really exceeded 40%.This result surprised many students.
The students learned about randomness from this demonstration and how easy it is to fake psychic abilities.In this example, the teacher is simply taking advantage of the fact that "people don't allow consecutive numbers to appear": people frequently switch back and forth between three numbers to create "randomness".In a truly random sequence, what is the probability that a 2 will appear after three 2s have already appeared?In fact, it is still 1/3, which is as likely as 1 or 3.But that's not the case for most people when it comes to generating random numbers.After even a small repeating segment, people often deliberately change the numbers in an attempt to create a "random" sequence.In this way, in our example, the teacher only needs to choose one of the other two numbers instead of the number that the student picked in the previous round before each round of guessing.例如，如果那个实验中的学生在上一轮说的数字是2，那么老师就会在下一轮的猜测中从1或3中任选一个。如果学生在上一轮说的数字是3，那么老师就会在下一轮的猜测中从1或2中任选一个。这样一个简单的把戏根本不需要什么通灵能力，就能保证猜中的概率高于33%——高于三个数字随机猜测的准确率。
人们总是认为，如果一个序列是随机的，那它就不应呈现有重复和某种模式。2005年关于“shuffle”模式（意即“随机播放”）的争议（Levy, 2005）就以一种幽默的方式证明了这一点。此模式将下载到iPod里的歌曲以随机的方式播放。很多用户抱怨说shuffle模式并不随机，因为他们经常听到同一专辑或流派的歌曲。当然，许多心理学家和统计学家在听到这类抱怨时只能暗自苦笑，因为他们了解我刚才提到的类似研究。科普作家史蒂芬·列维（Steven Levy, 2005）讲述了他经历过的类似事情。他的播放器似乎在起初的一个小时里偏爱史提利·丹（Steely Dan）的歌！但列维明智地接受了专家告诉他的事实：真正的随机序列，往往看起来不像是随机的，因为我们倾向于给所有事物都套上一种模式。在进行有关问题的研究后，列维总结道，“生命可能确实是随机的，iPod可能也是。但是，我们人类将永远有自己的套路和模式，只为让无序变得可控。即使真的存在缺陷，问题也不在shuffle，而在我们自己身上”（p.10）。
以上列举的涉及统计推理理解中出现的错误，仅为冰山一角，有可能阻碍人们正确理解心理学。有兴趣的读者可以阅读由吉洛维奇（Gilovich）、格里芬（Griffin）和卡尼曼（Kahneman）编写的《思维捷径和偏见：直觉判断心理学》（Heruistics and Biases: The Psychology of Intuitive Judgment，2002），它在这一方面提供了比较完整、详细的描述。
吉格瑞泽（Gigerenzer）的《计算的风险：如何察觉数字是在欺骗你》（Calculated Risks: How to Know When Numbers Deceive You，2002）对统计与概率做了很通俗的介绍（对没有受过任何数学训练的初学者尤其适用）。此外还有哈斯戴（Hastie）和达维（Dawe）的《不确定世界的理性选择》（Rational Choiceinan Uncertain World，2001）和拜农（Baron）的《思考和抉择》（Thinking and deciding，2000）以及尼克尔森（Nickersn）的《认知和几率：概率推理的心理学》（Cognition and Chance: The Psychology of Probabilistic Reasoning, 2004）。
本章中所讨论的概率思维具有巨大的实践意义。由于没有充分运用概率思维能力，医生们选择了效果欠佳的治疗方法（Baron, 1998; Dawes, 2001）；人们不能准确地评估环境风险（Margolis, 1996）；在法律程序中错误地使用信息（Foster & Huber, 1999; Lees-Haley, 1997）；政府和私人企业将数以百万计的资金用于不必要的项目（Arkes & Ayton, 1999）；动物不断被捕杀以至濒临灭绝（Baran, 1998; Dawkins, 1998）；对病人实施了不必要的手术（Dawes, 1988, pp.73-75）；有人做出了错误的财务判断，损失巨大（Belsky & Gilovich, 1999; Kahneman & Tversky, 2000; Zweig, 2001）。
当然，我们不可能在一个章节里全面地讨论统计推理。我们的目的就是想强调统计对于研究及理解心理学的重要性。不幸的是，当遇到统计信息时，我们还找不到一个放诸四海皆准的规则。功能化的推理技能不像科学思维中的其他部分那么容易获得，而是需要通过正规学习才能掌握。幸运的是，现在大多数综合大学和社区学院都提供了入门级的统计学课程，而且不需要大学程度的数学基础。在上这类课程之前，读者可以先阅读我刚才推荐的那些书。
尽管很多科学家都真诚地希望一般大众能够知悉和理解科学知识，但有时对一门学科的精通依赖于对某些信息的掌握，而对这些信息的掌握又只有通过正规的学习才能实现。如果说对一门学科的深入理解是一般外行人也能随便达到的，这是一种在学术上不负责任的态度。统计学和心理学就属于这一类学科。心理学家艾伦·班欧（Alan Boneau, 1990）调查了心理学教科书的作者，请他们列出学生在学习心理学时需要掌握的最重要的术语和概念。在所列出的术语及概念中，频率最高的100个词汇中有接近40%属于统计学和方法论的范畴。不精通统计和概率的人不可能成为称职的心理学家（Evans, 2005; Friedrich, Buday, & Kerr, 2000）。
不可否认，本书的一个目的就是要使心理学的研究能为广大读者所接受。然而，心理学进行理论建构所依靠的实证方法和技术与统计学是如此密不可分（这一点和其他很多领域一样，如经济学、社会学和遗传学），以至于没有一个人可以在对统计学毫无知晓的情况下精通心理学。因此，尽管这一章对于统计思维介绍得相当粗略，但它的主要目的是要凸显另外一个对于理解心理学至关重要的专业领域。
和大多数学科一样，心理学研究所得出的是概率式的结论——大多数情况下会发生，但并非任何情况下都发生。虽然这些结论并非是100%准确的（就像其他科学中的情况一样），但根据心理学研究及理论所做出的预测仍然是有用的。阻碍人们理解心理学研究的一个原因就是，人们很难用概率的术语来思考。在这一章里，我们讨论了几个相当精彩的研究实例，这些例子表明大多数人如何与概率推理背道而驰：当人们遇到具体的、具有鲜活性的证据时，就把概率信息抛到一边了。他们没有考虑到，较大的样本能够提供对于总体数值更为精确的估计。最后，人们表现出赌徒谬误（把原本无关的事件看成是有联系的）。赌徒谬误源于下一章将要讨论的一个更为普遍的倾向：未能认识到偶然性在决定结果时所起的作用。
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Notes: