Another Puzzling Picture
While still questioning the same witness from "The Puzzling Picture" (see the Resource linked below), the police showed her another photograph and asked her to again identify the pictured woman. The photo was fuzzy and the woman's face hard to discern. After gazing at the picture for several minutes, the witness finally answered: "I have neither sister nor brother, but my mother's daughter is this woman's mother's daughter." Once again, she refused to explain her meaning. Again, the police come to you, as a logician, for help: Assuming that the witness spoke truly, who is the woman in the picture?
Resource: The Puzzling Picture, 7/13/2009
Q: I have come upon a fallacy (or so I believe) that I cannot seem to categorize! It is a psychological fallacy whereby one values something differently based on context. For example, studies showed that 3 in 4 people will go to a store 10 blocks away in order to buy a clock radio for $25 instead of $50, but only 1 in 5 people will go to the same store to buy a computer for $1525 instead of $1550. My wife used this one on me the other day, when discussing laptops. She said when referring to a $1000 laptop purchase, "What's another $400?" I'm sure this applies to other scenarios not involving money ("We've come 500 miles; what's another 10 miles?"). Does this ring a bell to you?―Jim Craft
A: Ding-Dong! Yes, that's a familiar psychological phenomenon, and I've done the same sort of thing myself. I don't think that it fits any logical fallacies, but psychologist Thomas Gilovich has discussed it. He attributes it to "mental accounting", which is the tendency to treat some dollars as of different value than others because they fall in different "mental accounts". A typical example of this is "found money" or "house money", that is, money won in gambling or in a contest, which people are usually willing to spend more easily than money they've earned. Of course, a dollar is worth the same whether you worked for it or found it lying in the street.
The phenomenon that you've described is similar to mental accounting in that some dollars seem to be valued more than other dollars, but it differs in that mental accounts don't seem to be involved. Rather, whether it is worth driving across town to save money seems to be evaluated not on the absolute value of the savings, but on the percentage saved. However, if it's worth the extra expense in gas and time to drive across town to save $25, then it shouldn't matter what percentage of the item purchased you're saving.
This type of thinking may be related to another psychological phenomenon called "anchoring": people tend to make estimates based on numbers that they have recently been exposed to, even when the numbers are irrelevant to the estimate. So, perhaps we anchor our decision whether to go the extra mile to the price of the item to be purchased, even though that's irrelevant to whether it's worth the trip. What we should do instead is to make a rough estimate of the cost of the trip, and then make it if the savings exceed the cost.
One way that the current trillion dollar deficit worries me is that it may reset people's anchors: once we've passed the trillion mark, spending billions more may not seem like so much. "What's another $400 billion?" Anchoring may be a psychological mechanism that makes it easier and easier to go further and further into debt.
Source: Gary Belsky & Thomas Gilovich, Why Smart People Make Big Money Mistakes―and How to Correct Them: Lessons from the New Science of Behavioral Economics (1999), pp. 31-38.
Resource: Book Club: Nudge: Chapter 1: Biases and Blunders, 7/4/2008
How many albums did Michael Jackson sell? Was it 750 million, as USA Today reported? No one really knows, or if they do they're not saying, according to Carl "The Numbers Guy" Bialik. The number appeared in Wikipedia's biography of Jackson, and may have spread from there to the news media after his death. It's suspect for a couple of reasons:
- The source for the number appears to be Jackson's publicist, when Jackson was still alive and planning a comeback, who therefore had a motive for exaggeration.
- The number constitutes a sudden, unexplained jump in Jackson's sales from previous estimates of around 200 million.
Originally, Wikipedia cited the publicist's claim as its source for the number. For some unknown reason, the citation disappeared from the Jackson entry but the unsourced number is still there, which appears to be a violation of Wikipedia's sourcing policy. Moreover, its entry is currently "semi-protected", which prevents new users or those who lack Wikipedia accounts from editing it. So, even if I wanted to―which I don't―I am unable to change the unsourced claim, since I lack an account.
The publicist's original claim was that Jackson had sold 750 million "units"―whatever a "unit" is―but at the time of his death, Wikipedia attributed the number to album sales. Since news media usually don't give citations for such claims, it's impossible to know for sure whether they got it from Wikipedia, but I think it likely. As I write this, the Wikipedia article attributes "reported sales of over 750 million records" to Jackson, as opposed to just albums. That is doubtlessly closer to the truth, but is a "unit" a record or perhaps a song?
Because of Jackson's recent death, Wikipedia's entry is probably receiving a much larger number of visitors than previously. It is currently the third result produced by a Google search on "Michael Jackson". As a result, this pseudofact may be repeated in student papers, news articles, and perhaps even in future biographies. It appears that this is how a pseudofact is now propagated: a publicist makes an exaggerated claim, Wikipedia prints that claim without fact-checking it, then the news media pick up the claim and run with it―it's an encyclopedia, after all! By the time the editors at Wikipedia start backing away from the claim, the damage is done: it's now a "fact" that Jackson sold 750 million records.
- "Twelve facts about Michael Jackson", BBC News, 6/26/2009
- Carl Bialik, "Spun: The Off-the-Wall Accounting of Record Sales", The Numbers Guy, 7/15/2009
- Anthony Breznican & Gary Strauss, "The many faces of Michael Jackson", USA Today, 6/30/2009
In the Headlines
A trillion here, a trillion there, and pretty soon you're talking real money.―Everett Dirksen [adjusted for inflation]
Here are a couple of headlines:
Deficit reaches record $1 trillion and counting
Budget deficit tops $1 trillion for first time
Here are some questions raised―not begged―by these and other recent headlines:
- In what sense is a trillion dollar deficit a record?
It's a record in the same sense in which it's a first, that is, it's the first time the budget deficit has ever been as much as a trillion dollars. However, that raises the question whether it's a record when adjusted for inflation. Since inflation means that money loses value over time, a trillion dollars today is worth less than a trillion dollars fifty years ago. So, the fact that the current budget deficit is larger in dollars than previous deficits does not necessarily mean that it is larger in value. You might assume that reporters and editors routinely take inflation into account when they compose their stories and headlines, but you would be wrong. Rather, as I've often pointed out, they routinely fail to take inflation into account.
Adjusted for inflation, the highest previous deficit was as long ago as 1943―during World War 2―and that was less than half a trillion dollars (in fiscal year 2000 dollars). So, this is a real and significant record by over double the previous highest deficit.
However, in addition to inflation, the size of the economy has grown over time. So, that raises the question whether this is also the biggest deficit when compared to the size of the economy as a whole, which is usually measured by the gross domestic product (GDP). The 1943 deficit was about 30% of that year's GDP, whereas the estimated 2009 deficit―which is close to 2 trillion dollars―is only about 13% of the current GDP. In that sense, a trillion dollar deficit is far from a record.
- What's so special about a trillion?
Nothing, really, though people tend to like round numbers. This isn't irrational, since rounding off serves to avoid unnecessary and misleading overprecision, especially when dealing with very large numbers. But if human beings had eight fingers and toes instead of ten, we would presumably have an eight-based numbering system, and a one followed by twelve zeros would be a different number. Also, as we saw above, the record was broken when the deficit was less than half a trillion. So, why does the news media make a big deal about it passing a trillion? Perhaps because a trillion is such a large and unfamiliar number, and unfamiliar because it is so large.
- What is a trillion?
Most of us don't have a good sense of the difference in size between a million, a billion, and a trillion, and the deficit might as well be a "gazillion" for all it means to us; it's just a really big number. To put it in a little perspective, a trillion is a thousand billion, which is a thousand million; so a trillion is a million million. In other words, the deficit is the equivalent of a million millionaires. Alternatively, if the money were distributed equally to every man, woman, and child in the United States, it would amount to about three thousand dollars a head, or twelve thousand dollars per family of four. Wouldn't that stimulate the economy, or at least make the recession easier to bear for a lot of people?
- "Budget of the United States Government: Historical Tables Fiscal Year 2010", GPO Access, Table 1.3.
- "Deficit reaches record $1 trillion and counting", PLI SmartBrief, 7/14/2009
- Martin Crutsinger, "Budget deficit tops $1 trillion for first time", Associated Press, 7/13/2009
Resource: Douglas R. Hofstadter, "On Number Numbness", Metamagical Themas (1985), pp. 115-135. "Number numbness" is what Hofstadter calls the inability to appreciate the significance of very large and very small numbers.
Acknowledgment: Thanks to John Congdon for bringing this issue to my attention.
Update (7/18/2009): John adds:
There is one little detail that may complicate things further. If memory serves, "billion" and "trillion" mean different things in American and British English. As I understand it, (older) British English actually does have intermediate levels of "thousand million" and "thousand billion" between "million", "billion", and "trillion"―thus, "one trillion" for an American is ten to the 12th power (1,000,000,000,000), but ten to the 18th power (1,000,000,000,000,000,000) for an old-school Brit.
That's right, so it's possible that a British reader of an American headline might think that the U.S. budget deficit is a million times worse than it is, as if it weren't bad enough!
Source: Norman W. Schur, British English, A to Zed (1991), Appendix II-D.
The Puzzling Picture
While questioning a witness, the police showed her a photograph and asked her to identify the woman in the picture. "I have neither sister nor brother," she answered, "but this woman's mother is my mother's daughter." She then refused to say anything else. Confused, the police come to you for help: As a logician, can you identify the woman in the picture?
Check it Out
Given the inflation-adjusted record high price of gasoline a couple of years ago, it will probably be awhile before I can go back to complaining about reporters hyping high gas prices. So, for the time being I'll just have to gripe about journalists hyping supposedly record amounts of money earned by movies. Why are the box office earnings of movies not adjusted for inflation? It's obvious why the studios themselves don't do it, since they like to use the supposedly record box office of a movie as a selling point. But why do journalists go along with this puffery? The short answer is that the record-setting hype makes for good headlines. For the long answer, read the Slate article.
Source: Zachary Pincus-Roth, "Best Weekend Never", Slate, 7/6/2009
The Back of the Envelope
As an exercise in critical thinking, read the following passage from reporter John Stossel's book Myths, Lies and Downright Stupidity:
MYTH: DDT causes all kinds of cancers….
TRUTH: DDT saves lives.
Malaria will kill more than one thousand children before you finish reading this book. The chemical DDT is at the core of the problem―not the use of DDT, but the failure to use it because of media hysteria. In Uganda alone, said minister of health Jim Muhwezi, "We are losing between two million and three million people a year." Think of it: Millions die because the media gets it wrong.
Before proceeding, is there anything in the above passage that arouses your skepticism? If so, how would you go about checking it? Do so! Come back when you're done and read the rest of this entry; it'll still be here.
One critical thinking skill that everyone should learn is when and how to do "back of the envelope" calculations. Don't uncritically accept factual claims such as those in the above excerpt, but check them for plausibility. If they seem implausible based upon common sense or something else you already know, do some quick research. A few seconds on Wolfram Alpha or Google may be all that is required. Specifically, here are some questions about the excerpt:
- How long would it take to read Stossel's book?
- How many children die from malaria every year, according to Stossel? Is this a plausible number?
- If between 2 and 3 million Ugandans die of malaria a year, how many would die in all of Africa? In the whole world? Are these numbers plausible?
- Are these numbers consistent with one another?
If you haven't formulated any questions yourself and tried to answer them yet, then do so now, or try to answer the above questions.
Source: John Stossel, "Excerpt: 'Myths, Lies and Downright Stupidity'", ABC News, 11/17/2007
The Back of the Envelope: Here are the claims in the passage worth fact-checking:
- "Malaria will kill more than one thousand children before you finish reading this book."
Is this number plausible? This is a good candidate for a "back of the envelope calculation" (BOTEC). It's hard to tell when expressed in units of "reading Stossel's book", so try to convert it to a more usual unit, such as number of deaths per year. Of course, how long it takes to read the book will vary, but this is why a BOTEC is appropriate, since the goal is to come up with an approximation which can be checked for plausibility.
Stossel's book is approximately 300 pages in length. Let's suppose that it takes a minute on average to read a page: I tested a few pages of Stossel's book and it seems a good approximation. So, it should take about five hours to read the book. Let's be conservative and assume that a thousand children die of malaria in that five hour period. Then around 1.75 million children would die of malaria in a year. Since children are roughly a third of the total population, one could expect about 5.25 million total deaths per year. Of course, this depends on the definition of "child", and on whether malaria affects children and adults equally. If children are more likely to catch malaria or die from it than adults, then the total number of deaths will be lower. Let's be conservative and assume that most of the malaria deaths are of children.
Is 2 to 3 million deaths a year from malaria a plausible figure? You may have no idea―I didn't! However, the Ugandan Minister of Health quoted by Stossel claimed that 2 to 3 million people die per year in Uganda alone. If this were correct, one would expect the number of child deaths for the entire planet to be much higher than what Stossel says.
- "In Uganda alone, said minister of health Jim Muhwezi, 'We are losing between two million and three million people a year.'"
Is this number plausible? Who knows? Of course, you'd expect Uganda's own Minister of Health to know. However, Uganda is just one country in the continent of Africa. Unless malaria is especially prevalent in Uganda, one would expect that the total number of deaths worldwide would be at least an order of magnitude higher than in Uganda, and perhaps as much as two orders. That is, the total number of deaths would probably be at least 20 million worldwide, and perhaps more than 200 million. This sounds implausibly high to me. If malaria were really that bad a problem, wouldn't we hear more about it?
Another approach to assessing the plausibility of this figure is to compare it with the population of Uganda. Now, you probably don't know exactly what the population of Uganda is―I didn't! However, you may know that the population of the United States is around 300 million, and that Uganda's is probably considerably less than that. Three million deaths per year would be 1% of the population of the U.S.; so, it would be a higher percentage of the population of Uganda, perhaps much higher. Is it plausible that more than 1% of the population of Uganda is dying every year from malaria? If it were, wouldn't we hear more about it? Also, wouldn't Uganda be spraying DDT like crazy, no matter what environmentalists elsewhere think?
At this point, it's time to do a little research to answer the specific questions: What is the population of Uganda? How many people a year does malaria kill? If you haven't already done so, go check for yourself! Don't take my word for it!
According to the sources that I've checked, the population of Uganda in 2006, when the book was published, was around thirty million. So, if three million Ugandans died every year from malaria, that would be 10% of its population each year! Moreover, sources indicate that the worldwide mortality from malaria is about a million annually. So, unless these sources are wrong, it's impossible for Uganda to have as many malaria deaths as claimed by its Minister of Health.
Of course, this also means that the number of children who die of malaria while you read his book will be less than Stossel claims. Even if almost all of the million deaths are children, the number of child deaths in a five-hour period would be about half of what Stossel claimed. But isn't that enough? Or, rather, isn't that too many?
Here's the first myth in the excerpt from Stossel's book:
MYTH: The media will check it out and give you the objective truth.
TRUTH: Many in the media are scientifically clueless, and will scare you to death. We don't do it on purpose. We just want to give you facts. But the people who bring us story ideas are alarmed. Then we get alarmed, and eager to rush that news to you. We know that the scarier and more bizarre the story, the more likely it is that our bosses will give us more air time or a front-page slot. The scary story, justified or not, will get higher ratings and sell more papers. Fear sells. That's the reason for the insiders' joke about local newscasts: "If it bleeds, it leads." Also, raising alarms makes us feel important. If we bothered to keep digging until we found the better scientific experts, rather than the ones who send out press releases, we'd get the real story. But reporters rarely know whom to call.
I agree, and Stossel himself is a case in point, at least when it comes to malaria and DDT. This is why it is necessary to read critically, to know how and when to do a BOTEC to test claims for plausibility, and to be willing and able to check facts for yourself. Because you can't count on reporters to do it for you.
Stossel has done some good debunking work in the past, and elsewhere in his book. For instance, in the book excerpt, he rightly criticized the media tendency to hype every rise in gasoline prices as a "new record high" without adjusting for inflation. However, his misreporting on DDT and malaria calls into question everything else he reports, and thus undermines the good work he has done.
Since Stossel set out to debunk "myths, lies, and downright stupidity", I think that it's incumbent upon him to make an extra effort not to spread any myths or lies of his own. It would have taken only a few minutes for Stossel to have looked up the facts about malaria deaths and Uganda. Where are the editors at ABC News? Doesn't his publisher have fact-checkers? I don't know whether these mistakes were downright stupid, but they were stupid.
- "Malaria Fact Sheet", World Health Organization
- "Malaria Worldwide", Centers for Disease Control
- "Population", Uganda Bureau of Statistics
- "Uganda", CIA: The World Factbook
Update (7/8/2009): When I wrote the above, I had to guesstimate how long it would take to read Stossel's book because there was no explanation in the text as to how long that was supposed to be. The book has only a few endnotes, and the only note for the section concerning DDT and malaria did not explain how the number of child deaths while reading the book was estimated. However, in a later section on the "myth" of overpopulation (pp. 26-27), Stossel writes: "More than 125,000 babies will be born before you finish reading this book…." (P. 27) Stossel seems to like this type of comparison! The endnote for page 27 states: "According to WHO [The World Health Organization], 15,514 children are born every hour across the globe, and we estimate around 8 hours of reading-time for this book." (P. 286)
I have no problem with this estimate, though eight hours seems like a long reading time. However, it does give information for refiguring the BOTEC estimate of annual malaria deaths of children. Assuming a thousand children die of malaria in an eight-hour period, that means a little more than a million such deaths in a year's time. Given that most deaths from malaria are children, this is a fairly accurate estimate.
However, it does raise the question of how to square this information with the claim of two to three million deaths a year in Uganda. Stossel quoted the Ugandan Minister of Health as if he were a trustworthy authority, and did not challenge or raise any questions about his claim. How did Stossel arrive at the estimate of a thousand child deaths in an eight hour period? The obvious way would be to take the figure of a million malaria deaths a year, which are mostly children, together with the estimated reading time, and calculate the number of deaths in eight hours. If that's how Stossel came up with the number, how could he have not noticed the contradiction with the numbers given for Uganda?
Source: John Stossel, Myths, Lies, and Downright Stupidity: Get Out the Shovel―Why Everything You Know is Wrong (2006).
Addendum (8/16/2009): An old friend of The Fallacy Files, Michael Koplow, wrote to point out that the "Truth" that DDT saves lives, as cited by Stossel, is not contrary to the supposed "Myth" that it causes all kinds of cancers. In other words, it's possible that DDT saves lives by killing the mosquitoes that carry malaria, while at the same time causing cancer.
This is not the only "Myth" which Stossel's "Truth" does not deny. For instance, from the book excerpt linked to above:
MYTH: Divorce hurts women much more than men, and many men abandon their kids.
TRUTH: Both men and women suffer after divorce, and lots of men want to give more to their kids.
The supposed "myth" that divorce hurts women more than men is not refuted by the claim that both men and women suffer, since the "myth" does not deny that men suffer, rather it claims that women suffer more. Furthermore, the second part of the "myth", that "many" men abandon their children is not denied by the claim that "lots" of men wish to give more to them―"many" and "lots" are so vague that both claims could be true.
Solution to the Puzzling Picture: According to the witness, the woman in the photograph is her daughter. Given that the witness has no sisters―"I have neither sister nor brother" she says―the phrase "my mother's daughter" can only refer to herself. Therefore, her testimony is equivalent to: "this woman's mother is me", or "I am this woman's mother". Thus, the photograph is a picture of her daughter.
This puzzle is a variation on a traditional one in which a man looks at a portrait. I suspect that the traditional puzzle may be too well known to fool people today, so I simply changed the sexes. Here's the wording of the original puzzle as given by Raymond Smullyan:
A man was looking at a portrait. Someone asked him, "Whose picture are you looking at?" He replied: "Brothers and sisters have I none, but this man's father is my father's son." … Whose picture was the man looking at?
Since the man looking at the portrait has no brothers, "my father's son" must refer to he, himself. Thus, he is the father of the man in the portrait.
Smullyan remarks that many people mistakenly conclude that the picture is a portrait of the man looking at it. I can verify this to the extent that when I first encountered the puzzle when I was young, I had the same reaction. So, if you thought that the photograph in the puzzle was a picture of the witness herself, you're probably not alone!
Source: Raymond Smullyan, What is the Name of this Book? The Riddle of Dracula and Other Logical Puzzles (1978), pp. 7 & 14.
Solution to Another Puzzling Picture: As explained in "The Puzzling Picture", since the witness has no sisters the phrase "my mother's daughter" refers to her. So, what she said means: "I am this woman's mother's daughter" or "I am the daughter of this woman's mother." In other words, she claims that she and "this woman" have the same mother, which would seem to make her and "this woman" sisters. However, we already know that the witness has no sisters. Therefore, the woman in the photo can only be one person: the witness herself.