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Time to go shake the city and see what falls out. Bibliography of Map Projections, 2nd ed. Washington, Jamessam Betrouwbaarheid van de restoration?

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JerryAmofe Eryfug Jryfug Deneen Payout roulette - best online roulette for money - learn roulette. PaulUnosy CarlUnosy DenUnosy JudyUnosy JasonUnosy Only a perfect alibi saved the workman from an otherwise certain conviction.

Episodes such as this have undoubtedly happened in many courtrooms all over the world Gigerenzer On a formal level, a probability of 2.

Even in a Bayesian setting with certain apriori-probabilities for guilt and innocence, one finds that a probability of 2.

And from the frequentist perspective, which is more common in forensic science, it is nonsense to assign a probability to either the null or to the alternative hypothesis.

Still, Students and, remarkably, teachers of statistics often misread the meaning of a statistical test of significance.

The test was supposed to detect a possible treatment effect based on a control group and a treatment group.

The subjects were asked to comment upon the following six statements all of which are false. They were told in advance that several or perhaps none of the statements were correct.

Ironically, one finds that this misconception is perpetuated in many textbooks. Additional examples are collected in Gigerenzer , chap.

On the German market, there is Wyss , p. Does this imply that its logical equivalent "If not A then not B" has the same probability attached to it?

Setting aside the fact that John Paul II has not been randomly selected from among all human beings, one finds that this argument again reflects the confusions that result from "conditioning with conditional events".

Or in terms of rules of logic: If the statement "If human then not Pope" holds most of the times, one cannot infer, but sometimes does, that its logical equivalent "If Pope then not human" likewise holds most of the times.

Strange as it may seem, this form of reasoning has even made its way into the pages of respectable journals.

For instance, it was used by Leslie to prove that doom is near the "doomesday argument", see also Schrage In this case the argument went: 1 If mankind is going to survive for a long time, then all human beings born so far, including myself, are only a small 8 proportion of all human beings that will ever be born i.

Conditional probabilities and favorable events The tendency to confuse conditioning and conditional events is often reinforced by an inclination to conclude that a conditional probability that is seen as "large" implies that the reverse conditional probability is also "large".

The confusion occurs in various contexts and is possibly the most frequent logical error that is found in the interpretation of statistical information.

These examples can easily be extended. Most of them result from unintentionally misreading the statistical evidence.

When there are cherished stereotypes to conserve, such as the German tourist bullying his fellowvacationers, or women somehow lost in space, perhaps some intentional neglect of logic may have played a role as well.

Also, not all of the above statements are necessarily false. It might, for instance, well be true that when men and women drivers are given a chance to enter a one-waystreet the wrong way, more women than men will actually do so, but the survey by Bild simply counted wrongly entering cars and this is certainly no proof of their claim.

For example, what if there were no men on the street at that time of the day? And in the case of the Swiss skiing resort, where almost all foreign tourists came from Germany, the attribution of abnormal dangerous behavior to this class of visitors is clearly wrong.

In words: When A is favorable to B, knowing that A obtains increases the probability of B, and knowing that A does not obtain decreases the probability of B.

The British Home Office nevertheless once did so in its call for more attention to domestic violence Cowdry Evidently not.

While it is perfectly fine to investigate the causes and mechanics of domestic violence, 11 there is no evidence that the private home is a particularly dangerous environment even though, as The Times mourns, "assaults Favorableness and Simpson's Paradox Another avenue through which the attribute of favorableness can be incorrectly attached to conditioning events is Simpson's paradox Blyth , which in our context asserts that it is possible that B is favorable to A when C holds, B is favorable to A when C does not hold, yet overall, B is unfavorable to A.

One instance where Simpson's paradox to be precise: the refusal to take account of Simpson's paradox has been deliberately used to mislead the public is the debate on the causes of cancer in Germany.

However, as Table 1 shows, among women, the probability of dying from cancer has actually decreased for young and old alike! Similar results hold for men.

Still, the willful disregard of the most important explanatory variable "age" has turned the overall increase in cancer deaths into a potent propaganda tool.

If B is favorable to A, then by a simple calculation B is unfavorable to A. However, B an still be favorable to subsets of A.

This is also known as Kaigh's paradox. In words: If knowing that B has occured makes some other event A more probable, it makes the complementary event A less probable.

However, we cannot infer that subsets of A have likewise become less probable. Schucany , Table 1 gives a hypothetical example where Kaigh's paradox is used to misrepresent the facts.

Suppose a firm hires out of applicants among which are Black, are Hispanic and White. The selection rate for Hispanics is thus less than that for Whites.

A German newspaper quoted in Swoboda , p. In fact, a glance at any statistical almanac shows that quite the opposite is true.

This time the confusion is spread by Alan Dershowitz, a renowned Harvard Law professor who advised the O. Simpson defense team. Instead, the relevant probability is that of a man murdering his partner given that he battered her and that she was murdered: P K battered and murdered.

It must of course not be confused with the probability that O. Simpson is guilty; a jury must take into account much more evidence than battering.

How to make the sources of confusion disapear Almost all fallacies discussed above can be attributed to the unwarranted application of what we have elsewhere called "fast and frugal heuristics" Gigerenzer Heuristics are simple rules that exploit evolved mental capacities, as well as structures of environments.

When applied in an environment for which they were designed, heuristics often work well, commonly outperforming more complicated optimizing models.

Nevertheless, when applied in an unsuitable environment, they can easily mislead. When a heuristic misleads, it is not always the heuristic that is to blame.

More often than not, it is the structure of the environment that does not fit Hoffrage et al. We have seen examples of this here with what has been called the base-rate fallacy Borgida and Brekke In fact, this environmental change underlies most of the misleading arguments with conditional probabilities.

Consider for instance the question "What is the probability that a woman with a positive mammography result actually has breast cancer?

Conditional probabilities: The probability that a woman has breast cancer is 0. Take, for example, a woman who has a positive result.

What is the probability that she actually has breast cancer? Natural frequencies: Our data tells us that 8 out of every women have breast cancer.

Of these 8 women with breast cancer 7 will have a positive result on mammography. Of the women who do not have breast cancer some 70 will still have a positive mammogram.

Take, for example, a sample of women who have positive mammograms. How many of these women actually have breast cancer?

Apart from rounding, the information is the same in both of these summaries, but with natural frequencies the message comes through much more clearly.

Natural frequencies correspond to the way humans have encountered statistical information during most of their history.

They are called "natural" because, unlike conditional probabilities or relative frequencies, on each occurence the numerical qualities in our summary refer to the same class of observations.

For instance, the natural frequencies "7 women" with a positive mammogram and cancer and "70 women" with a positive mammogram and no breast cancer both refer to the same class of women.

This switch of reference class easily confuses the minds of both doctors and patients. Half the doctors received the information in conditional probabilities and half received the data as expressed by natural frequencies.

In contrast, most of the doctors who were given natural frequencies gave the correct answer or were close to it. Simply converting the information into natural frequencies was enough to turn much of the doctor's innumeracy into insight.

Presenting information in natural frequencies is therefore a simple and effective mind tool to reduce the confusion resulting from conditional probabilities.

Figure 1 17 References J.

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For example, what if there were no men on the street at that time of the day? And in the case of the Swiss skiing resort, where almost all foreign tourists came from Germany, the attribution of abnormal dangerous behavior to this class of visitors is clearly wrong.

In words: When A is favorable to B, knowing that A obtains increases the probability of B, and knowing that A does not obtain decreases the probability of B.

The British Home Office nevertheless once did so in its call for more attention to domestic violence Cowdry Evidently not. While it is perfectly fine to investigate the causes and mechanics of domestic violence, 11 there is no evidence that the private home is a particularly dangerous environment even though, as The Times mourns, "assaults Favorableness and Simpson's Paradox Another avenue through which the attribute of favorableness can be incorrectly attached to conditioning events is Simpson's paradox Blyth , which in our context asserts that it is possible that B is favorable to A when C holds, B is favorable to A when C does not hold, yet overall, B is unfavorable to A.

One instance where Simpson's paradox to be precise: the refusal to take account of Simpson's paradox has been deliberately used to mislead the public is the debate on the causes of cancer in Germany.

However, as Table 1 shows, among women, the probability of dying from cancer has actually decreased for young and old alike!

Similar results hold for men. Still, the willful disregard of the most important explanatory variable "age" has turned the overall increase in cancer deaths into a potent propaganda tool.

If B is favorable to A, then by a simple calculation B is unfavorable to A. However, B an still be favorable to subsets of A. This is also known as Kaigh's paradox.

In words: If knowing that B has occured makes some other event A more probable, it makes the complementary event A less probable.

However, we cannot infer that subsets of A have likewise become less probable. Schucany , Table 1 gives a hypothetical example where Kaigh's paradox is used to misrepresent the facts.

Suppose a firm hires out of applicants among which are Black, are Hispanic and White. The selection rate for Hispanics is thus less than that for Whites.

A German newspaper quoted in Swoboda , p. In fact, a glance at any statistical almanac shows that quite the opposite is true.

This time the confusion is spread by Alan Dershowitz, a renowned Harvard Law professor who advised the O. Simpson defense team. Instead, the relevant probability is that of a man murdering his partner given that he battered her and that she was murdered: P K battered and murdered.

It must of course not be confused with the probability that O. Simpson is guilty; a jury must take into account much more evidence than battering.

How to make the sources of confusion disapear Almost all fallacies discussed above can be attributed to the unwarranted application of what we have elsewhere called "fast and frugal heuristics" Gigerenzer Heuristics are simple rules that exploit evolved mental capacities, as well as structures of environments.

When applied in an environment for which they were designed, heuristics often work well, commonly outperforming more complicated optimizing models.

Nevertheless, when applied in an unsuitable environment, they can easily mislead. When a heuristic misleads, it is not always the heuristic that is to blame.

More often than not, it is the structure of the environment that does not fit Hoffrage et al. We have seen examples of this here with what has been called the base-rate fallacy Borgida and Brekke In fact, this environmental change underlies most of the misleading arguments with conditional probabilities.

Consider for instance the question "What is the probability that a woman with a positive mammography result actually has breast cancer?

Conditional probabilities: The probability that a woman has breast cancer is 0. Take, for example, a woman who has a positive result. What is the probability that she actually has breast cancer?

Natural frequencies: Our data tells us that 8 out of every women have breast cancer. Of these 8 women with breast cancer 7 will have a positive result on mammography.

Of the women who do not have breast cancer some 70 will still have a positive mammogram. Take, for example, a sample of women who have positive mammograms.

How many of these women actually have breast cancer? Apart from rounding, the information is the same in both of these summaries, but with natural frequencies the message comes through much more clearly.

Natural frequencies correspond to the way humans have encountered statistical information during most of their history. They are called "natural" because, unlike conditional probabilities or relative frequencies, on each occurence the numerical qualities in our summary refer to the same class of observations.

For instance, the natural frequencies "7 women" with a positive mammogram and cancer and "70 women" with a positive mammogram and no breast cancer both refer to the same class of women.

This switch of reference class easily confuses the minds of both doctors and patients. Half the doctors received the information in conditional probabilities and half received the data as expressed by natural frequencies.

In contrast, most of the doctors who were given natural frequencies gave the correct answer or were close to it. Simply converting the information into natural frequencies was enough to turn much of the doctor's innumeracy into insight.

Presenting information in natural frequencies is therefore a simple and effective mind tool to reduce the confusion resulting from conditional probabilities.

Figure 1 17 References J. Bar-Hillel and R. Falk : "Some teasers concerning conditional probabilities. Beck-Bornholdt and H. Dubben : "Is the Pope an alien?

Hamburg Rowohlt. Blyth : "Simpson's paradox and mutually favorable events. Borgida and N. Brekke : "The base rate fallacy in attribution and prediction.

Harvey, W. Ickes and R. Kidd Eds. Chung : "On mutually favorable events. Cowdry : "Half women murder victims killed by husbands or lovers.

Edwards : "Is the Pope an alien? Feller : An Introduction to probability theory and its applications. New York Oxford University Press.

Gigerenzer : Calculated risks: How to know when numbers deceive you. London Penguin. Gigerenzer and A. Edwards : "Simple tools for understanding risks: From innumeracy to insight.

Gigerenzer : "Fast and frugal heuristics: The tools of Bounded Rationality. Koehler and N. Harvey eds.

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