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One instance of consciously exploiting such computational errors in order to cheat the public is a game of cards called "Three Cards in a Hat" which used to be offered to innocent passers-by at country fairs in Germany and elsewhere.

One card is red on both sides, one is white on both sides, and the other is red on one side and white on the other. The cheat draws one card blindly, and shows, for example, a red face up.

The cheat then offers a wager of 10 Deutschmarks that the hidden side is also red. The passer-by is assumed to argue like this: "The card is not the white-white one.

Therefore, its hidden side is either red or white. The example therefore boils down to an incorrect enumeration of simple events in a Laplace-experiment in the subpopulation composed of the remaining possibilities.

Bourgnet from March 2, , reprinted in Leibniz , pp. A sum of 11, so he argued, can be obtained by adding 5 and 6, and sum of 12 by adding 6 and 6.

It did not occur to him that there are two equally probable ways of adding 5 and 6, but only one way to obtain 6 and 6. Prominent textbook examples are the paradox of the second ace or the problem of the second boy see for instance Bar-Hillel and Falk , not to mention the famous car-and-goat puzzle, also called the Monty-Hall-problem, which has engendered an enormous literature of its own.

These puzzles are mainly of interest as mathematical curiosities, and they are rarely used for statistical manipulation.

We shall not dwell on them in detail here, but they serve to point out what many consumers of statistical information are ill prepared to master.

Confusing conditional and conditioning events German medical doctors with an average of 14 years of professional experience were asked to imagine using a certain test to screen for colorectal cancer.

The prevalence of this type of cancer was 0. The doctors were asked: "What is the probability that someone who tests positive actually has colorectal cancer?

The most common fault was to confuse the conditional probability of cancer, given the test is positive, with the conditional probability that the test is positive, given that the individual has cancer.

An analogous error also occurs when people are asked to interpret the result of a statistical test of significance, and sometimes there are disastrous consequences.

A forensic expert correctly computed a probability of only 0. From this figure the expert then derived a probability of 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

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Time to go shake the city and see what falls out. Bibliography of Map Projections, 2nd ed. Washington, Unfortunately, it was. Although the term conditional probability does not appear once in Huff's remarkable book, it is clear that many other examples of statistical abuse that he has discovered can be rephrased in terms of conditional probabilities.

Below we survey various ways in which such reasoning can be misleading, and we provide some fresh examples. We also show that the potential for confusion is easily reduced by abandoning the conventional, "singularly clumsy terminology" of conditional probabilities in favor of presentation of information in terms of natural frequencies.

Fallacies in enumeration One class of errors involving conditional probabilities comprises outright mistakes in computing them in the first place. One instance of consciously exploiting such computational errors in order to cheat the public is a game of cards called "Three Cards in a Hat" which used to be offered to innocent passers-by at country fairs in Germany and elsewhere.

One card is red on both sides, one is white on both sides, and the other is red on one side and white on the other. The cheat draws one card blindly, and shows, for example, a red face up.

The cheat then offers a wager of 10 Deutschmarks that the hidden side is also red. The passer-by is assumed to argue like this: "The card is not the white-white one.

Therefore, its hidden side is either red or white. The example therefore boils down to an incorrect enumeration of simple events in a Laplace-experiment in the subpopulation composed of the remaining possibilities.

Bourgnet from March 2, , reprinted in Leibniz , pp. A sum of 11, so he argued, can be obtained by adding 5 and 6, and sum of 12 by adding 6 and 6.

It did not occur to him that there are two equally probable ways of adding 5 and 6, but only one way to obtain 6 and 6.

Prominent textbook examples are the paradox of the second ace or the problem of the second boy see for instance Bar-Hillel and Falk , not to mention the famous car-and-goat puzzle, also called the Monty-Hall-problem, which has engendered an enormous literature of its own.

These puzzles are mainly of interest as mathematical curiosities, and they are rarely used for statistical manipulation. We shall not dwell on them in detail here, but they serve to point out what many consumers of statistical information are ill prepared to master.

Confusing conditional and conditioning events German medical doctors with an average of 14 years of professional experience were asked to imagine using a certain test to screen for colorectal cancer.

The prevalence of this type of cancer was 0. The doctors were asked: "What is the probability that someone who tests positive actually has colorectal cancer?

The most common fault was to confuse the conditional probability of cancer, given the test is positive, with the conditional probability that the test is positive, given that the individual has cancer.

An analogous error also occurs when people are asked to interpret the result of a statistical test of significance, and sometimes there are disastrous consequences.

A forensic expert correctly computed a probability of only 0. From this figure the expert then derived a probability of 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.

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