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Quantitative Literacy and the Humanities

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Probability and the Humanities: Pitfalls and Questions

Even in cases where we do not have precise numbers for calculating probabilities or expected values, students recognize that it is useful to understand some of the logical reasoning that undergirds probabilistic thinking—and some of the common pitfalls.  The humanities help us to think through and anticipate these probabilistic pitfalls.

The answers to many of these questions rely on both mathematical and humanistic thinking.  As such, they may not have "correct" answers, although some answers will be more convincing and involve more rigor, rationality and reflection.

Assuming two events are independent, when they are actually dependent

If two events are totally independent of each other, and we know the probability of them each happening, then we simply multiply the probabilities to get the probability that they will both happen.  But if two events are related—one causes the other, or they are both caused by a third event—then the probability of both events is much, much higher.

Example: The housing crisis.  Before 2008, many analysts believed that bundling mortgages together made them a less risky bet.  What was the probability mistake behind this belief?  Answer!

Humanists can analyze an event from multiple perspectives, articulate multiple outcomes of the same cause, and explain how events develop in a dynamic and interconnected process.  They develop a broad frame of reference that allows them to see relationships between two seemingly disconnected events.

Assuming two events are dependent, when they are actually independent

Sometimes we see connections or patterns where there aren't any.  When "streaks" to happen, we assume that they will continue.  But we should recall the reversion to the mean: unusual events are likely to be followed by results consistent with the long-term average.

Example: The “gambler’s fallacy.”  If a roulette wheel lands on red five times in a row, what is the probability that the next spin will be red?  Answer!

The same skills that allow humanists to see connections can help them to decide whether a connection truly exists.  They are skeptical observers of patterns.  And they are well aware of the role that contingency plays in the unfolding of events.

Discounting the unlikely event

We tend to believe that an unlikely event will never occur.  This statistical pitfall leads us to be unprepared for unlikely events, or to assume that the occurrence of such events was due to a particular malice.  But given enough time or enough trials, unlikely events do in fact occur: A 99 percent probability of success may not be good enough, if 1% of the time a catastrophe will strike.  After enough time, you will hit that 1%. 

Here's another way to look at it: “Yes, the probability that five people in the same school or church or workplace will contract the same rare form of leukemia may be one in a million, but there are millions of schools and churches and workplaces" (Wheelan, Naked Statistics, 103-104).  Any one outcome may be very unlikely, but in the long run, one unlikely outcome or another is very likely to occur.

Example: The "prosecutor's fallacy." A DNA test has a one in a million chance of making a false match.  DNA taken from a crime scene is put through a database of over a million random people, and locates a match.  Is that person guilty?  Answer! 

Humanists have developed a broad frame of reference that includes a long time frame, millions of people and many thousands of villages, towns, and institutions.  They can imagine a variety of unlikely events that might occur over the aggregate.  And, they appreciate the importance of contingency in the unfolding of events.

Ignoring the conditions

The probability of an outcome changes given that certain conditions apply. 

Example 1: Breast cancer.  In the breast cancer problem presented elsewhere, we saw that a woman with a positive result had only an eight percent chance of having the disease.  This is based on a random screening for breast cancer.  Can you imagine a condition under which the probability would be different?  Answer!

Example 2: The "prosecutor's fallacy" revisited.  In the DNA database presented elsewhere, we saw that a DNA match located through a large database could very likely be a coincidence.  Can you imagine a condition under which the match would be more likely?  Answer!

Humanists are accustomed to considering different scenarios, conditions and contingencies.  They know that the outcome of events can change based on these conditions.

Using the wrong time frame

The probability that an event will happen in a one-time attempt is different from the probability over the course of a year, or over the course of a lifetime.  Similarly, probabilities based on one moment in time can overlook differences in groups over a longer period of time.

Example 1: Breast cancer revisited.  The probability that a woman will develop breast cancer today is not the same as this year or over the course of her lifetime.

Example 2: Financial models.  “Financial models based on the 1990s are not good predictors about long-term risk" (Wheelan, Naked Statistics, 99).

Example 3: The "half of sufferers" fallacy.  Here's a scenario from John Paulos that explains how the time frame pitfall might mistakenly yield the headline HALF OF SUFFERERS ARE LONG TERM.  Answer!

Humanists are adept at adjusting the time frame to encompass a lifetime, a century, a millennium or even millions of years.

Mistaking a precise model for an accurate model

Just because a model yields precise numbers does not mean it is based on good data or on a good model. 

Example: A weather forecast that predicts a high temperature of 68.4 degrees is precise, but if it is based on a broken thermometer or an outdated model, it is not very accurate.

Humanists are accustomed to evaluating the processes by which answers are derived.  They can assess models because they are adept at looking for patterns and for causal relationships.  They are also accustomed to evaluating evidence, so they can assess the value of the data.  Even if they lack the technical expertise to evaluate a particular model or means of data collection, humanists are curious skeptics when presented with a precise finding.

Forgetting that calculation needs interpretation

Calculation may yield a precise answer, but it does not always tell us what to do with that information.  We need to consider many factors, including ethics, one's appetite for risk, and one's desire for the expected payoff.

Example 1: Breast cancer revisited (again).  If universal screening for women leads to false positive tests, is it a good policy to screen all women for breast cancer?  What are the benefits and drawbacks of such a policy?

Example 2: Insurance.  What is the purpose of insurance? When is it prudent to buy insurance (or a warranty)?  Answer!

Example 3: Probability or profiling?  Is it ethical to use statistics that rely on data about personal characteristics—such as race, nationality or religion—in order to fight crime or stop terrorism? At what point does a probability-based model turn into profiling?

Example 4: Betting on coin flips.  As told by John Paulos: “Two men bet on a series of coin flips, agreeing that the first to win six flips will be awarded the $64,000 stake. The game, however, is interrupted after only eight flips, with the first man leading five to three. The $64,000 question is, How should the pot be divided?"  (Paulos, Mathematician, 47-48)  Answer!

The humanities allow us to imaginatively enter into a variety of situations, and therefore to broaden our ability to interpret calculations and make recommendations based on those calculations.  The humanities expand empathy, broaden one's frame of reference, engage in ethics, and provide a range of human experiences with risk and payoff. 

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