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In Chapter 6, Wheelan explains the various flawed assumptions that often accompany probabilities. He starts off talking about VaR and the economy in 2008, which made false assumptions in its models predicting economic security that ultimately led to the recession and stock market crash. He then continues to explain many different ways that people make incorrect assumptions when talking about probability. They may believe events to be independent when there was some unknown factor influencing it, or they may be assumed to be connected when they were independent. Someone might consider a cluster to be statistically significant, when it really was simply luck. He also brings up biases and other ways that people may discriminate when thinking about statistics that can lead them to a false conclusion.

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On page 98, Wheelan says when talking about VaR, "Very little attention was devoted to the 'tail risk,' the small risk (named for the tail of the distribution) of some catastrophic outcome." This passage strikes me as interesting because it highlights the comfort we as humans feel in seeing 99% chance that ____ won't happen, despite the fact that it means in 1 out of 100 times something catastrophic will happen. This lack of respect for the possible catastrophic outcome ends up creating unwarranted feelings of security without having ways to prevent catastrophes.

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On page 99: "The Wall Street quants made three fundamental errors. First, they confused precision with accuracy... Second, the estimates of the underlying probabilities were wrong... Third, firms neglected their 'tail risk'" (99). This passage highlights how these seemingly simple mistakes have massive ramifications even/especially for the high-prestige statisticians of our society; these mistakes are just silly goofs that high schoolers might make in their math tests. Again, the preconceptions, confidence, and misunderstandings of people using statistics are the danger of using statistics itself.

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In Chapter 6, Wheelan tackles common flawed assumptions about various scenarios involving probability. He opens with a very hard hitting example: the 2008 financial crisis in which the ways the quantitative analysts on Wall Street 1) mistook precision for accuracy, 2) relied on incorrect underlying probability estimates, and 3) neglected 1% disaster scenario. This example leads to a series of explanations describing faulty misconceptions. First, assuming events are independent leads to incorrect probability calculations if, in fact, those outcomes may be correlated somehow. Two jet engines may fail a small amount of times, but because they can fail together when they actually do, we cannot assume that the engines fail completely independently. Next, not understanding when two outcomes actually are independent can lead people to believe that outcomes of the past can influence future or present probability (like in gambling), but this is not the case in many scenarios. Third, clusters of similar/same outcomes occur, but depending on the size of an observed population, this could simply be chance. Fourth, prosecutor's fallacy means finding outcomes that match an unlikely outcome can not be conclusive. Fifth, extremes in a trend usually trend back to the mean. Lastly, statistical discrimination can certainly be useful for some individuals/groups, but is it ethical?

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On page 100, the passage "Probability offers a powerful and useful set of tools–many of which can e employed correctly to understand the world or incorrectly to wreak havoc on it" resonated with me because it I find it interesting how powerful statistics, and especially probability, can be. It also implies that probability itself is not dangerous, and can be used as a very useful tool, but when people make mistakes and use it incorrectly, the damage can be awful depending on the situation.

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In Chapter 6 of Naked Statistics, Wheelan describes the numerous ways that people can use probability incorrectly, and the consequences of this. He starts the chapter with an extreme example of the stock market crash in 2008. He described how Wall Street had three fundamental errors when evaluating the risks of investment: confusing precision with accuracy, the inaccuracy of the underlying probabilities, and the "tail risk," or the small chance that the negative outcome would happen. Wheelan hen goes on to describe common errors in evaluating probabilities. First, assuming linked events are independent. By not considering the connection between the two events, the probability can be greatly and inaccurately skewed. Next, not knowing when events are independent. Similar to the last error, considering a connection that does not exist will wrongly affect the probability, and the assessment of the risk. Wheelan describes that clusters happen. Just because something has a low probability of happening does not mean that it won’t happen multiple times in the same area, especially with a large data set. Wheelan then describes what he calls the prosecutor’s fallacy, which is when the context of the statistics is completely ignored, leading to a false claim that seems to be supported by the statistics. He also discusses reversion to the mean, when an abnormal or exceptional event happens, and you expect it to continue to happen because it already happened once. However, after this event, it is likely to return to the mean, and not have the same outcomes as the previous, exceptional event. Finally, Wheelan describes statistical discrimination, such as discrimination based on gender, which can create a false sense of accuracy. Wheelan concludes the chapter by stating that probability is a tool that can be very useful in determining if something is more or less likely, but it can be used incorrectly, yielding inaccurate results.

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On page 96, the passage "In other words, 99 times out of 100 the firm would not lose more than $13 million on a particular trading position; 1 time out of 100, it would" resonated with me because often times people assume that because a certain variable has a high chance of happening, it is sure to happen. On the contrary, however small the minority percentage is, the experimental and theoretical probability will be different because the small chance still has a chance of occurring. This chance could still occur 3 times out of 10 in an experiment although the theoretical probability predicted a 1 percent chance that the event would occur.

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In chapter 6 of Naked Statistics, it talks about how probability is not perfect or always correct in predicting whether an event will occur. He describes that there is a difference between probability situations where the variables are dependent on each other, such as a weather pattern, and situations such as a coin toss that the variables are unrelated. He describes that a mistake people make is thinking certain events rely on each other and relate causation to correlation which are not the same thing. He describes that a common misconception is that if there is a majority chance of one thing happening, people rule out the other. This happens during elections, where if one candidate has a 70 percent chance and one has a 30, people say that the first will for sure win. This is not true, and experimental and theoretical probabilities can prove themselves different. In the presidential scenario, the second candidate could still win a majority of the times if the experiment was done multiple times due to chance.

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The final part about statistical discrimination is particularly interesting. On page 108, Wheelan questions, “Is it okay to discriminate if the data tells us that we’ll be right far more often than we’re wrong?” The dilemma is difficult to answer definitively and it makes me consider the more complex factors that go into the creation of racially profiling statistics. Could the reason for greater numbers of Hispanic drug smugglers be due to the lack of education available in their area? Could the imposing limitations of America on minorities be the reason for such a high statistic? I think the underlying reasons behind the statistics are important to consider as well.

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Chapter 6 of Naked Statistics is all about how people can incorrectly use probability and the risks associated even when used correctly. The chapter begins with how a model that evaluated the probability of financial losses in the stock market caused a financial crisis. There were three primary flaws in the model he highlights. One, the model was precise, not necessarily accurate, leading to a false sense of security. Two, the data was based on the last few decades, which is not a valid prediction when it comes to the irregular and fickle nature of the stock market. Lastly, the model did not consider the scope of the unlikely event actually occurring. Going into more detail, there are common mistakes when it comes to probability. Assuming events are independent when they are in fact dependent and vice versa, can lead to incorrect calculations. There is also the possibility of an unlikely outcome to happen, especially in large sample size. If the outliers are individually selected out of a large data set, it can lead to false and unjustified conclusions. Another misconception is that deterioration or improvement is caused by some outside factor when it could be due to the “law of large numbers” and regression to the mean. The last idea Wheelan brings up is the issues that arise even when we use statistics correctly. Especially when it comes to religion or ethnicity, it could still be wrong to use statistical models as a justification to treat others differently.

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On page 100, Wheelan says "probability doesn't make mistakes; people using probability make mistakes." I find this quote important because it fits into the broader theme of the book which is that even though statistics is powerful and useful, people can weaponize and abuse them. It also fits into the theme that math doesn't lie, people do.

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Chapter 6 details the common errors and their outcomes of using probability. At the core of the chapter, Wheelan describes the common errors people use when using probability. The first error is assuming events are independent when they're not, such as engine failure on a plan. The second is the opposite which is when someone assumes events are dependent when they're not, such gambling or flipping a coin. Third is when clusters occur, which is when a series of unlikely outcomes occur just do to chance, which can make people think there are more factors at play. Fourth is the prosecutor's fallacy, which states that you cannot determine something simply because it is likely; you have to look at the context and explore more potential reasons for such event. Next is reversion to the mean which states that after one or a series of unlikely events occur, the outcomes will shift back to what is most probable. Lastly, he talks about statistical discrimination, when applying statistics and probability to policies can have social and discriminatory implications.

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In the sixth chapter of "Naked Statistics," Charles Wheelan explores the many problems with probability (as chapter's title explicitly states). He opens this discussion by describing "one of the most irresponsible uses of statistics in recent memory," the events leading to and including the 2008 financial crisis (95). In doing so, he introduces the event's three fundamental errors: 1) the confusion between precision and accuracy, 2) the inaccuracy of the estimates of the underlying probabilities (based on the past),  and 3) the negligence of the "tail risk" (did not account for the catastrophic implications of the most unlikely event to occur). As Wheelan defines the chapter's purpose of eliciting the most common probability-related errors, misunderstandings, and ethical dilemmas, he makes the core claim of the chapter: "Probability doesn't make mistakes; people using probability make mistakes" (100). In other words, it is not the math that is incorrect, but rather how people use and present that math. Wheelan then offers the six fundamental errors of probability. The first is assuming events are independent when they are not. An independent event is when the first event does not affect the outcome of the next (as in flipping a coin). Many people often mistake dependent events, however, which is when a certain outcome makes a similar outcome more likely in future trials, for independent events,  having catastrophic and sometimes even fatal results. The second error is in not understanding when events ARE independent. In direct contrast to the first described error, this one (also referred to as "the gambler's fallacy) entail individuals who treat independent events as dependent. For instance, no matter how many times you consecutively flip heads on a coin,  the probability of again flipping heads is still 1/2. The third error is misattributing a factor other than chance to a cluster of highly improbable cases. The fourth is  called the "prosecutor's fallacy" and occurs when the context surrounding statistical evidence is neglected (common in court hence the name). The fifth error is the misunderstanding of the phenomenon known as "reversion to the mean." It is when outcomes that are more consistent with the long-term average/mean follow an anomalously good stretch/winning streak. The final error Wheelan discusses is statistical discrimination, which begs the question of when it is appropriate to act on the basis of what probability says is likely and often involves our moral obligations. Overall, statistics is a powerful tool, but it also a dangerous weapon. Wheelan urges us to be cautious and responsible in how we use and present statistics.

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On page 109, the passage stating, "for all the elegance and precision of probability, there is no substitute for thinking about calculations we are doing and why we are doing them" stood out to me because it connects back to our discussion of precision versus accuracy and of the ethics involved in statistical reasoning (109). in a previous chapter, Wheelan defined precision as the "exactitude with which we can express something," while accuracy is "a measure of weather a figure is broadly consistent with the truth. The problem with these ideas in statistics that we can use precision to mask inaccuracies, resulting in a false sense of faith/certainty, whether given deliberately or unintentionally. As Wheelan reinforces in this sixth chapter, ethics and moral responsibility does hand-in-hand with this dilemma. He argues that there is certain weight involved with both how and why we use our calculated statistics. Even if the math itself is correct ("precise"), Wheelan cautions us to consider how we understand and/or present such information, being careful to uphold ethical/moral obligations we have to fellow citizens of our world.

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On page 97, while explaining the faults of the VaR model for the stock market, Wheelan compares the VaR to a "faulty speedometer, which is arguably worse than no speedometer at all. If you place too much faith in the broken speedometer, you will be oblivious to other signs that your speed is unsafe. In contrast, if there is no speedometer at all, you have no choice but to look around for clues as to how fast you are really going." I really like this simile because it easily illustrates where people can go wrong in using probability. Although probability is very powerful, we cannot rely on it fully. Additionally, we must be wary of models using probability and consider their faults, because as Wheelan demonstrates, if it is like a "faulty speedometer," you are completely mislead. Therefore, while using probability we must be aware of its drawbacks and critical of what is actually telling us, remembering that probability can only tell us the likelihood, not certainty, of something occurring.

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Chapter 6 focuses on the problems with statistics, specifically where people can make mistakes while using statistics. Wheelan introduces this idea with the example of the 2008 financial crisis. He points out that the VaR gave people a false sense of security by guaranteeing a 99% chance of working and a 1% chance of error; however, unlikely things occur and "tail risk" needs to be considered. Additionally, the Wall Street quants confused precision and accuracy and used incorrect estimates for the underlying probability. This example leads Wheelan into an organized description of the mistakes people using probability make. First, he explains the danger in assuming two events are independent when they are actually dependent or failing to see when events actually are independent. He then points out that although something might seem too random to occur, it can and clusters of data can happen. He also describes the danger of prosecutor's fallacy, you cannot determine something just because it is extremely likely, you must also consider the alternative. Wheelan moves on the describe how data tends to demonstrate patterns of reversion to the mean, so an outlier is generally followed by outcomes consistent with the average. Finally, he introduces the idea of statistical discrimination and the complications of applying statistics to real-life scenarios because it will have social implications. In conclusion, this chapter explores the places where people applying/using probability need to be cautious because it is a powerful tool, but it reveals multiple places for people to create misleading conclusions.

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The quote I found interesting was, "in financial markets (unlike beer tasting), the future does not necessarily look like the past. There was no intellectual justification for assuming that the market movements from 1980 to 2005 were the best predictor of market movements after 2005" (97). I thought this was interesting because it reveals an underlying assumptions that I think most statistics are based on that things will stay the same. If you really had no idea how things would change in the future, I don't think you could make any predictions. Most of the time you can estimate limits to how much things will change, but even then you still can't be certain as the Great Recession showed.

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Chapter 6 is about the problems that arise when we rely on statistics to make decisions. Most of these problems seem to be based around either humans assuming things which are not said by the statistical model and/or models not considering enough costs to accurately predict the benefits over a large number of costs. For example, most people think that getting the same result from flipping a coin several times predicts that the next result will be different, even though probability doesn't know anything about the forces and angular speed which really predict the result of the coin flip. Also, while stopping all people will certain qualities may have the expected benefit of reducing drug trafficking, that doesn't consider the cost of many innocent people being harassed by the police.

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In Chapter 6, Wheelan discusses the problems with probability (as shown in the chapter's title), specifically the common errors, misunderstandings, and ethical dilemmas that arise due to the use, or misuse, of statistics. The first error he explains is "assuming events are independent when the are not" (100), which can subsequently falsely create a sense of confidence/security in an outcome because it assumes independence rather than dependence of variables to creating said outcome. The second error is "not understanding when events are independent" (102). In reverse to the last error, this one occurs when you assume different events are dependent, rather than independent of each other, which can culminate into what's known as the "gambler's fallacy." The third misunderstanding is that "clusters happen" (103). When a cluster of an unlikely outcome occurs in one location, we generally assume that there's another motivating factor that resulted in the cluster, when, in reality, it could've happened by chance. The fourth misunderstanding is called the prosecutor's fallacy. This fallacy occurs when a person doesn't take into consideration the context surrounding the statistical evidence, which can lead to false conclusions or highlight other explanations to the outcome that occurred. The fifth misunderstanding is called "the reversion to the mean (or regression to the mean)" (105). The reversion to the mean occurs when a person, after having an outcome that is either super good or super bad, reverts back to their long-term average. Finally, Wheelan highlighted the controversial relationship between statistics and discrimination. He draws on the question of how statistics and social constructions interact, such as "is it okay to discriminate if the data tell us that we'll be right far more often than we're wrong?" because it forces us to consider the application of statistics in our modern world (108).

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On page 98-99, I found the quote "'The greatest risks are never the ones you can see and measure, but the ones you can't see and therefore can never measure. The ones that seem so far outside the boundary of normal probability that you can't imagine they could happen in your lifetime—even though, of course, they do happen, more often than you care to realize'" to be very striking in that it nicely articulates the importance of taking into account the risk factor to any event/decision you make. I think, in general, we sometimes assume the worst-case scenario won't happen to us, and this notion can be problematic since bad things do happen. I think the author of this quote, Nicholas Taleb, nicely words the importance of being realistic and recognizing the potentiality of big problems. I also think this quote highlights the limitations of statistics as well in that sometimes some risks can't be measured or foreseen.