Here are some of the other examples I have been getting from complete strangers around the world. If you do not want your examples posted here E-Mail me at paul@xocxoc.com. Personal Info, redundant info, and compliments on how much people liked my site (thanks!) have been edited. Stuff I wrote appears in italics.
Topics:
Suggested Mistakes and Examples
Criticisms and Examples (by subject)
The Infamous "Lets Make a Deal" Problem
Suggested Mistakes and Examples
A humorous Statistical Rash:
A) The Japanese eat very little fat and suffer fewer heart attacks than the
British or Americans.
B) On the other hand, the French eat a lot of fat and also suffer fewer heart
attacks than the British or Americans.
C) The Japanese drink very little red wine and suffer fewer heart attacks than
the British or Americans.
D) The Italians drink excessive amounts of red wine, and also suffer fewer heart
attacks than the British or Americans
E) Conclusion: Eat & drink what you like. It's speaking English that kills
you. - Author Unknown
Paul, did you hear about the enormous sampling mistake made by the group
(headed by Jack Valenti, I think) that opposes underage drinking? They said that
25 percent of all drinking was now done by underage drinkers. They then had to
retract the information and said it was actually 11 percent or something. But
they also said that now underage girls drink every bit as much as underage boys.
I wouldn't be surprised if there was an error there, too. - Submitted by Jack
Shea
I have not had time to research this, but the study I am interested in seeing
is how much has raising the drinking age to 21 affected the consumption of
alcohol. From what I can tell, raising the drinking age to 21 has done nothing
except make criminals out of 90% of the 18, 19, and 20 year olds.
On NPR a few weeks ago, I heard a most interesting statistic. A researcher at some university had arrived at
the astounding conclusion that the more male children you have in a family, the more likely it is that one of them
will grow up to become a homosexual.
Why stop there? I would argue that this also increases the likelihood that one of them will grow up to be president,
or a used car salesman, or be hit by a meteorite, or win the lottery, or contract syphilis, or rob a bank, or die
while ice-fishing, or... - Submitted by Scott Hancock
I have noticed a mathematical quirk, often exploited by the media, which is not mentioned in your site. When dealing with small numbers, minor fluctuations can be made to appear far more significant than they really are. For example - Assume that inflation last year was 1%. Now assume that executive salaries grew at an annual rate of 2.5%. One could accurately report that "Executive salaries grew at a rate equal to 250% of the rate of inflation." However, it would be more useful (and more honest) to report that executive salaries grew at a real rate of just 1.5%. - Submitted by Paul Hugel
I'm a big fan of your page Paul. I revisit it often to figure which of the various hoaxes I'm being subjected
to today as I read articles or listen to speakers. I'm writing a book about Y2K and trying to explain the kinds
of hysteria raising techniques being employed. I've figured out my own explanation of how the hoax works but I
can't find this particular mistake on your page. Therefore, here it is.
THE DOUBLE STANDARD SHELL GAME
When engineers design things, they typically assume the worst. If a design can withstand the worst of everything
it ought to be good enough. A large risk for designers is overlooking something. That risk is what led Edward A.
Murphy to pose his famous law. "Anything that can fail, will." To designers, Murphy's law is an admonition
to avoid focusing on some hazards at the risk of overlooking others.
When we use things already designed, we use an operational standard of risk presumption. "Nothing will go
wrong that we can't handle." This is the attitude you must take to, for example, board an airplane. If you
go ahead and board without believing the operational premise, you're a fool. Positive thinking is so important
in operational situations that we even allow it to be enforced by law. It can be criminal to speculate loudly about
all the things that can go wrong on a flight, or to make jokes about bombs, while in the boarding area. Like shouting
fire in a crowded theater, this is one of the few prior restraints on speech that we tolerate.
Few people are aware that these two standards of risk presumption exist. They are therefore vulnerable to the double
standard shell game. First someone tells a story about a hazard, say GPS systems failing in August 1999. Then they
mention that the things at risk can be critical. Then they project images of crashes, death and suffering; clearly
something from the operational context. The implication is that the listener should connect the hazard, with the
Murphy's law presumption, to the operational expectation.
Fear mongers pull the switch on us frequently. Think of the typical TV news magazine program. Each program probably
pulls this trick two or three times per hour, week after week, year after year. Many listeners may suspect that
it may be hype, but they never do figure out exactly how they are being conned.
ASSUMING THE WORST CASE IS ALWAYS SAFER
Murphy's law is not a joke, it is a credo for design engineers to follow."If anything can fail, it will."
In system design, this means prepare for the worst. It is a standard for safety design that is beyond debate. Can
it be misapplied? In the nuclear industry, the occupation of many analysts for their entire careers, has been simulation
of the maximum credible accident. The theory is that if a design can be found that successfully withstands this
accident, then it should be trusted for all less severe accidents also. Therefore, the maximum credible accident
came to be called the Design Basis Accident, or DBA for short.
The philosophy for calculating DBA is simple. Whenever a question is raised, assume the worst. All uncertainties
are eliminated from consideration by assuming the worst. That must be correct, right? If we assume the worst, and
things come out OK, then they must be OK for less than the worst. Right? In very complex processes, like nuclear
reactors, there are thousands and thousands of uncertain factors and thus an equal number of worst case assumptions.
This was the credo for DBA calculations for more than 40 years. Finally, in the 1980s, the thinking shifted. The
DBA calculations with their thousands of "conservative" assumptions were so far removed from experience
that they can not be verified. Because they deal with fantasies, common sense sanity checks could not be applied.
Such fanciful DBA calculations could contain numerous mistakes that nobody would notice because the whole scenario
was so unrealistic.
The standards were changed to make the expected outcome the center of gravity. The validity of expected outcome
models can be verified by comparison to reality. Conservative assumptions are superimposed on these models, but
they are now required to keep one foot anchored in reality. - Submitted by Dick Mills
How can you explain that, based on the principle that I have a father and a mother who also had the same, my
ancestors were more than the entire population of the earth today (2,4,8,16,etc... becomes some 30 billions at
the... 40th generation (at 50 years per generation that's year one of the Christianity) I never received a satisfactory
answer so I called it the Delor
Paradox and hope you will help me. - Submitted by Tom Delor
Actually, I believe the 'official' term is inbreeding. I like to think of this in reverse. I was into genealogy
for a while and was shocked that I am a direct descendent of William the Conqueror (who lived in the 11th century,
25 generations back), just like the British royal family. But, come to think of it, at least 99.9% of all people
with some British ancestry can claim the same thing. Everyone of at least some European blood are descendants of
Charles the Great (a.k.a. Charlemagne who lived in the 9th century, 33 generations back in my case) whose descendants
intermarried with every royal line in Europe. And while the royal families of the Middle East pride themselves
in being descendants of Mohammed (7th century founder of Islam), it is a good bet that more than half of the world
population is as well. And, virtually the whole planet descended from Abraham (around 16th century BC) whose descendants
are proclaimed to be the "Chosen People of God", I know I feel blessed. So, the next time someone thinks
I am crazy, I can just claim my parents were cousins.
In news coverage we often hear of "the fastest-growing sport/hobby/whatever in America." If I invent a sport today and teach it to my brother the number of participants has doubled in a single day and become a candidate for "fastest-growing." When we hear this phrase, we should assume that something of very limited interest has just begun to attract a few followers. The more popular a sport or hobby is the less likely it is to rank high on the list of "fastest-growing" ones. - Submitted by Paul Brians
You've probably heard of the birthday thing. In case you haven't, compute the likelihood of any two people
in a small group of being born on the same day. Sounds like high odds, but the answer is surprising. - Submitted
by Jan R. Schwenk
To be absolutely certain of finding two people that have the same birthday, you need 367 people. To be 50% sure
you only need 27 people. I'll leave it to the reader to figure out why.
I'd like to offer the following as an example of useless mathematics.
Statistics that suggest nothing: Frequent news stories report that changes in life style (lower fat intake
for instance) can reduce your death rate from heart attacks by, say, 20%. That number has no value whatsoever.
There is almost never reference to what the original rate was leaving us with no sense of scale. But even
if they reported that, for instance, the number dying per 1000 men over 60 was reduced by X from 600 to 600 -X
it would still have little value. Are they suggesting that if people improve their eating habits 20% fewer
would die from any cause? The only other possibility is that the 20% savings in lives will permit you to
die from other causes. So the statistic is not only useless, it's not very comforting. It leaves you
wondering just what it will be that does you in. - Submitted by Harry Puncec
I read an article in the Deseret News about a ninth grader in Idaho who did a study on critical thinking.
He asked 50 of his classmates if they would be willing to sign a petition calling for the elimination of dihydrogen
monoxide. His argument was that it carries bacteria that cause disease, in the air it causes people to sweat,
it is a major cause of soil erosion and it causes death. Of the 50 classmates he asked, 34 signed the petition,
15 refused to
sign the petition, and 1 realized that he was trying to ban water.
Regarding the statistical correlation argument. The surgeon general claims that smoking causes lung cancer.
The tobacco industry disputes that claim. Who is right. I say that the tobacco industry is correct.
Smoking only increases a persons risk of developing lung cancer. If smoking caused lung cancer all smokers
would get lung cancer and this is not the case. I'm not defending the tobacco industry but a lot of scientists
and others seem to forget that statistical correlation does not prove cause and effect. - Submitted by Stacey E.
Haws
The Dihydrogen Monoxide facts sheet can be found at http://extlab1.entnem.ufl.edu/IH8PCs/humor/water.htm.
According to an article in the Skeptical Inquirer (Jan 1998 pg. 13) the survey was done by Nathan Zohner, 14, of
Eagle Rock Junior High. Economist James K Glassman wants to coin the term "Zohnerism" defined as the
use of a true fact used to mislead a scientifically and mathematically ignorant public. A term worthy of
the Glossary. Also, I am not sure how serious you are being with your smoking does not cause lung cancer
argument, by the same reasoning, guns do not kill people because not everyone who is shot with a gun dies.
Since there is a proven statistical correlation between smoking and lung cancer (no one denies this), then either
1.) smoking causes lung cancer or 2.) lung cancer causes smoking or 3.) people with a high risk of lung cancer
also have a high risk of enjoying tobacco.
In the Wizard of Oz, when the wise man gave the lion (?) the brain, the lion still didn't get the brains of a math teacher. The Lion proudly quoted, " The sum of the squares of the sides of an isosceles triangle is equal to the square of the other side". (it should be a right triangle, although the right triangle could be isosceles so I gave him half marks for effort.) - Submitted by R Hammond
My favorite mistake of this kind is the one that involves "atoms" or, in this century, "particles"(as in 'top quark'). Specifically, the argument that two particles could ever be, or proven to be, absolutely identical. - Submitted by Greta or Dad Joe Piecora
Please consider adding the following obvious and ubiquitous "mistake," to wit, the nine-digit zip
code. Mathematical hyperbole.The post office has been pushing their nine-digit zip code for years. Any one with
training in arithmetic, i.e. anyone with a fourth-grade education, can figure out that with nine digits one could
assign no less than three (3) unique zip codes to every man, woman and child in the United States, which has a
population of less than 333,333,333 people. An ex-girlfriend once sent me a letter from Boston to Philadelphia
using only my nine-digit zip code, with no other information on the envelope. It arrived in a timely fashion. -
Submitted by Ted Nason
I have not thought of that before, the only flaw in the argument is that many urban zip codes often contain
more than 10,000 households in it. Social Security numbers will probably never run out, being made of 9 digits
as well. Credit card numbers are 16 digits, the first digit indicates the type of card (Visa = 4), followed by
3 digits indicating the bank issuing the card. The last two digits are a checksum indicating automatically a good
or bad number, The remaining 10 digits are unique numbers. - Xocxoc
How about advertisers who claim "a 500% growth in sales in the last year" Great - so they sold one item at the beginning of the year, and this year they sold 6 ! (or better yet - they sold 0 and still sell 0 ; 0 + 500% = 0 ! ) - Submitted by Colin Kalis
I noticed, while driving my parents `95 GMC Suburban, that the tachometer is labeled rpm/1000 rather than rpmX1000.
I enjoy cars and appreciate precision in thinking but I couldn't help to think to myself, "...and I driving
this machine designed by morons?" - Submitted by Pat Busby
Another Reader Responds: In physics and chemistry, it is standard to label axes on graphs "resistance/Ohms"
or "initial rate/mol s^-1" where the units are divided into the quantities so that units aren't needed
on the graph where they'd
just muddle the numbers. Likewise, it makes perfect sense for the tachometer to say "rpm/1000"
because if the dial is at 6, that means it is 6000 rpm/1000 = 6 on the dial. rpmX1000 would technically mean
that the true rpm = number on dial/1000, which is obviously false unless you're driving a 1967 Dodge Dart, in which
case .006 rpm is pretty damn impressive. And yes, I know I have too much free time. - Submitted by Matt Hallanger
A few readers have commented on this (sorry I cannot give credit, but I had to format my drive and I lost all
my old E-Mail), whether or not the rpm/1000 is correct depends on if you think the number the tachometer is pointing
to the divisor or the answer of the equation. If the value on the tach is x then x = rpm/1000 or rpm = x times
1000. I myself drive a Geo Metro which does not have a tachometer (apparently the makers were too embarrassed with
its results) except in the form of a light that turns on when the rpm is high enough to warrant switching gears.
You may want to add my favorite,"accumulating savings" similar to adding percentages:
A friend of mine came to work sweaty and out of breath. Upon asking him why he told me that he saved $1 by running
behind the bus. I told him "tomorrow, why not save $5 by running behind a cab". How many times have you
heard "the more you spend, the more you save!!!". Yogi Bara would say "The more you spend, the more
you spend". - Submitted by Thomas Dobroth
Two days ago, in a well respected Israeli paper, there was an interview with the CEO of American Greeting Card
Co. The opening paragraph and some other places in the text stated that in the U.S. 35 million greeting cards/year
are sold, for a total worth of 8 billion dollars. Obviously nonsense, even if you can't guess that the average
American probably gets more than a seventh of a card a year, anybody should be able to calculate that the average
greeting card doesn't cost $228.50 apiece.
Coffee consumption - commonly stated that per capita coffee consumption is highest in Sweden. Well, of course,
how much coffee do little kids drink? Birth rates in Sweden are among the lowest in the world, and per capita consumption
is probably much more dependent on the percentage of adults in the total population than it does with how much
they really drink.
Personal histories - Ever notice in how many news interviews they state that someone is x years old, then relate
his job or other history since age 21, and you end up with a lot more years than 21-x? - Submitted by Len Kaplan
In the June 1997 Consumer Reports (CR), under the heading "How safe are sport-utility vehicles," CR wrote: "More Americans are now dying in crashes involving a car and a light truck (such as an SUV) than in crashes involving two cars, even though cars outnumber light trucks two to one and are involved in more crashes." This statement is hardly surprising. Everything else being equal, if there are twice as many cars as light trucks, the probability that a (two vehicle) crash involves two cars is 4 in 9; the probability that it involves a car and a light truck is 4 in 9; and the probability that it involves two light trucks is 1 in 9. Also, the probability that the crash will involve a car is 8 in 9; and the probability that it will involve a light truck is 5 in 9. With this simple model, car-car crashes have the same probability as car-light truck crashes, even though crashes involving cars outnumber those involving light trucks 8 to 5. Furthermore, it is highly improbable that the number of car-car crashes exactly equals the number of car-light truck crashes, so with this model there is a fifty-fifty chance that there are more car-light truck crashes than car-car crashes (and vice versa). Therefore, the statement has a fifty-fifty chance of being correct even if the safety of cars and light trucks is identical. - Submitted by Don Eckhardt
Conspiratorial Coincidence
You have a nice example here. Alas that no one will be convinced by it. There's a lot of theorizing about why people want to disbelieve in true coincidence, but that they do is plain. - Submitted by Tom Zaslavsky
Drama Digits
Do you know of the late mathematician Paul Erdos? He was famous for real math, but also for his sayings, such as that he was 2.5 billion years old. When he was born, the Earth was 2 billion years old. When he was old, the Earth was 4.5 billion years old. QED - Submitted by Tom Zaslavsky
Graph Errors
The graphing of SAT scores vs. education expenditures is of particular interest. The public outcry regarding
declining SAT scores is misplaced, since the percentages of students that take the SAT has not remained constant.
In 1960, the first year on the graph, only approximately 2% of graduating seniors (the top students) took the SAT.
Now around 30% of all students take the test--a substantial difference. A more accurate picture would be to compare
the scores of the top 2% of students to the 1960 scores.
The debates over whether increased spending produces better students, or whether one curricular focus is more successful
than another, are stabbing at minutia. The best predictor of student success is not a direct result of anything
that happens within the school, but rather is related to the number of books in the home, the parents' education
level, and family income. So all the talk about "good" and "bad" schools, public vs. private,
integrated math vs. traditional, or whole language vs. phonics gives policy makers and the public a lot of fodder
for debate, but does little to improve the lives of children. But, then, any improvement that would really matter
would require a total shift to family-friendly economic policies and workplace, and this country is not willing
to make that investment. - Submitted by Peggy Barber
Actually, there has been just such a study of SAT scores, comparisons of the top 100,000 scores from 1960 to
the top 100,000 from 1990 shows a decrease of scores even among the top students, but not quite as dramatic as
the drop in scores overall, which you point out are not that valid. As for the rest of the stuff about the debate
of spending vs. family life, I agree with you totally. - Xocxoc
If you want a good example of a ZOOM GRAPH take a look at the daily stock market report onhttp://www.stockfind.newsalert.com/ - submitted by Chuck Neuenschwander, Dakota County Tech College
The "Kevin Bacon" Game
The so-called "Kevin Bacon game" is, perhaps unintentionally, a recent version of the "Erdos
number", well known to mathematicians. You have Erdos number 1 if you coauthored a paper with Erdos, 2 if
you coathored with a coauthor (but not with Erdos, of course), and so on. This number has been talked about for
possibly more years than Kevin Bacon has been alive. I was rather offended when the NY Times last summer called
it the mathematical version of the Kevin Bacon game. Such ignorance!
But speaking of "Six Degrees of Separation", that seems to be a minor mathematical mistake. It should
be seven degrees of separation, if the quote provided by the IMDb is accurate--and if one agrees that two related
people have one degree of separation, since 0 degrees of separation means being identical--but one could disagree
about this. - Submitted by Tom Zaslavsky
I am a little guilty of the same thing, I was familiar with the Erdos Number before I was familiar with the
Kevin Bacon game. I read about it from an old Martin Gardner article that was published before Kevin Bacon has
made his first movie. - Xocxoc
Law of Averages Thinking
I got to the bit on "The Law of Averages thinking", which says that even after tossing ten heads the
odds on the11th throw of heads coming up again is still 1:1. I'm curious 'cos when I was in college we were told
(by one of the good lecturers!) that in a case like that, where after ten throws you'd expect roughly 5 heads and
5 tails, but instead you have 10 heads, the odds of heads coming up again are higher 'cos the previous 10 heads
is pointing to some biasing factor. Is that complete bull? - Submitted by Sacha
Yes and no, assuming that the coin is "fair" it is true that the odds are going to be 1:1 regardless
of the past history. Coins have no way of remembering if they are succeeding or failing. If you were to flip a
coin 100 times and write down H for heads and T for tails, you will be surprised how often you get a long string
of H's or T's. Unlike coins, people do have memory. Basketball star Dan Majerle of the Miami Heat makes a little
more than 50% of his shots.The odds of his making a shot are about 1:1. And, like a coin he has hot streaks and
cold streaks. But, unlike a coin he can remember. If he misses a few shots in a row, his confidence is down and
he usually misses. When he is on a hot streak, he cannot miss because he has confidence in himself. - Xocxoc
Another reader Responds: Actually, the theory of "hot streaks" may itself be a fallacy, despite
player and spectator expectations. The New Scientist ran an article sometime in the past few months by a statistician
who analyzed game and player statistics , I think for tennis, and found that when a player had been doing well
up to a certain fixed point (say, half-time), the chances that they would score better than their season average
during the rest of the game were only fifty-fifty. That is, after half a game on a "hot streak", they
were no hotter during the rest of the game. Similarly for people who had been winning all their games up to a certain
point.
Yes, people do have memories, but they have false memories. A player on a hot streak may be more likely to *believe*
that he can't miss and to discount the misses he makes and focus on the scores. Then a "hot streak" is
a kind of Cancer Cluster + Astrology Amnesia. - Submitted by Eliza Sachs
I'd add the following: "It is different to ask in advance of any coin-tossing, What is the likelihood that it will come up heads ten times in a row? The answer is, tiny. Eleven times in a row? Even tinier. But once we actually start tossing the coin, each toss has an even chance of coming up heads or tails." - Submitted by Steve Tiger
Your explanation of Law Of Averages thinking points out yet another problem found confusing quite often:
while the Probability of rolling a 3 with a fair die is 1 : 6 or 1/6 the ODDS of rolling a 3
would be 1 : 5 or the ODDS against it would be 5 : 1 Similarly, the Probability of flipping Heads is
1 : 2 while the ODDS of flipping Heads is 1 : 1 - submitted by Chuck Neuenschwander, Dakota County Tech College
I have been meaning to add a topic of mathematical definition errors. The difference between ODDS and PROBABILITY
being a popular one. Another one is AVERAGE, which can be either MEAN average, MEDIAN average, or MODE average.
Most of the time it is MEAN, but if the MEDIAN is a more favorable of an out come one can legitimately use that
instead.
Logical Fallacies
You asked for better examples -- for the Fallacy of Ambiguity, consider:
(i) Bread crumbs are better than nothing, (ii) Nothing is better than steak, therefore (iii) Bread crumbs are better
than steak.
Submitted by Lance
You write; Fallacy of Emotion - an appeal to popular passions such as pity, fear, brute force, snob appeal,
vanity, or some other emotion. Example: You would look sexy behind the wheel of this new $50,000 sports car.
I disagree that it's a fallacy. Not only do I think I'd look sexy in a new Viper but I think a blond chick driving
a viper would look a lot sexier than if she was driving a VW bus. :-) - Submitted by Jim Lewis
It is a fallacy because a $50,000 is at least $30,000 more than you have to spend to get around town. If you
goal is to look sexy, then spend away. (This from someone who drives an ugly blue '91 Geo Metro) :-]
On your "reader-submitted examples" page, one fellow submitted what he thought was another example of the fallacy of ambiguity. His example actually is a specimen of equivocation. "Nothing" is used in two separate premises with two distinct meanings, which clearly is equivocation. Ambiguity is the use of a word or phrase that has an unclear meaning. For excellent explanations and examples of both, see T. Edward Dahmer's "Attacking Faulty Reasoning" (ISBN is 0-534-21750-8) on pages 64 for ambiguity and 62 for equivocation. - Submitted by Luke
Number Inflation
All I see in your examples is that two sources are reporting greatly disparate numbers. You are assigning the value "reality" or "actual" to one of them. This is extremely subjective and not based on mathematics. Under reporting is just as likely as over reporting. Different groups often use different criteria in their counting. There is also evidenced a presumption that government sources are more objective and therefore more reliable than advocacy sources. This is not necessarily the case. All anyone can conclude when confronting widely disparate numbers from different sources is that none of them may be accurate. To determine reality requires further investigation, analysis, and criticism. - Submitted by Tom Malloy
Of course, one must be careful that the numbers one is using for comparison do not DEFLATE the subject in question. In the spousal abuse comparison, for instance, the FBI statistics in question might be the UCR which is notoriously inaccurate for a variety of fundamental reasons, the NCVS which is more accurate in some ways, a study commissioned by the FBI, or numbers which the Feds pulled out of a hat to show how much the situation had improved since they first started publishing reports. Or gotten worse..- Submitted by Todd Ellner
I don't entirely approve of your examples. The undoubted fact that advocates tend to pump up the numbers in attendance at a march or rally doesn't mean that official, e.g., police figures are "actual", or that they are more--or less--accurate than the advocates'. I attended a rally once at which I personally counted all the people: there were about 1250. (This is not a hard number to count, especially since we were marching in a circle.) The police estimate was 2000. The organizers proudly announced this and said (roughly) "and so we must have FIVE thousand!". On another occasion I was in a perfect position to watch all the marchers at one of the anti-Vietnam war marches and count them as they went by (fall, 1969, it must have been, although I don't recall exactly). I counted, if memory serves (unfortunately, I didn't write it down) about 500,000. Of course I don't mean this to be more precise than within a few 10,000's. The official estimate was, if I recall, 300,000 or 400,000. The organizers claimed 500,000 (as best I can recall), which was quite close. - Submitted by Tom Zaslavsky
Number Numbness
Well, maybe you believe politicians who worry about spending (like our governor Pataki) $40,000 too much on a program to help first drug offenders straighten up while there's a several-billion-dollar budget surplus are suffering from number numbness. I'm more cynical. But you're right to think this is important. Maybe the constituents are the numb ones, since they don't seem to catch the fallacy. More likely it's that old standby, believing what you want to believe. Don't blame innumeracy for ills it isn't responsible for! - Submitted by Tom Zaslavsky
Opportunity Cost
With respect to your high school diploma being worth $280,000, I think that you make a numerical mistake that
is often made in the economic context. You fail to correct for the present value of the future income.
That is, if you are positing that for each of the next forty years a person with a high school diploma will earn
$7000 more than a counterpart without the high school diploma, the value of the high school diploma is the present
value of a 40 year, $7000 per year income stream. In order to determine what this is, you need to choose an appropriate
discount rate. The present value fallacy most often rears its head in the mortgage context. People say things
like, "Do you realize that if you pay your home off over 30 years, you are actually paying three times the
value of the home or more??" Of course, you are not actually paying three times the value of the home.
Instead, you are exchanging a stream of payments in the future from you to the bank, for a lump sum payment from
the bank to you now. Of course, this may already be taken into account in your employment example. You may
mean that this year you will make $7000 more, and that next year you will make $7000 plus inflation more (so, e.g.,
if inflation is 3%, next year the high school grad will make $7210 more). In other words, your example may
posit that the wage different grows at the discount rate in which case you can just add the numbers. If this
is the case though, you should make it clear. - Submitted by Professor Douglas R. Cole, University of Oklahoma
Law Center
Good info on how inflation works, as for the high school diploma example, I figure the two inflations cancel one
another out - Xocxoc
Percentage Pumping Formula
I made up this example to show the nonsensical 200% discount: "If the same $100 item were on sale for $33, for a savings of $67, the real discount would be 67% while the pumped-up discount would be 200%, which makes no sense at all." - Submitted by Steve Tiger
Is this percentage pumping , or what?
Front page article in Hastings,MN Gazette stated: "Since inmates at the Dakota County Jail have had to pay
for over-the-counter medications, there has been a 500 percent decrease
in demand." I called and asked the staff writer what that meant. He did not know but simply stated that
that number had been given him by another agency. That agency had no comment. The article goes on, by the way to
state that "criminal suspects and defendants are charged a $10 booking fee when they enter the jail, under
a new law passed in 1997" They had supposedly collect over $2,000 in the first month. So, what do they
do if you refuse to pay it? Throw you in jail? But then, they couldn't, could they, if you refused to pay
the booking fee. - submitted by Chuck Neuenschwander, Dakota County Tech College
Percentage pumping is usable, or at least present, when you look at percentage of increase/decrease as done within counting. Some areas compare the change to the original, some to the final value. Example: a cost of $75 marked up to $100. If we compare the increase to the original, this is a 33 1/3% increase. But business will, in some fields, compare the increase to the final amount, calling this a 25% increase. They officially call these Markup based on Cost and Markup based on Selling Price. - submitted by Chuck Neuenschwander, Dakota County Tech College
Practically Zero Probability
Sorry, you made the mistake! NO statistician coming across such a number would EVER call it zero. In one billion tries (which may be reasonable; I don't even know what Powerball is) it is reasonably likely, although not probable, that someone will win. (I make this statement based on a crude estimate that may be way off. To make an intelligent estimate I'd have to know how Powerball is played. But my correction, stands regardless.) NO probability "MUST" be considered zero. Where did you get this! - Submitted by Tom Zaslavsky
Rare Scares and Scaremongering
One thing that I have noticed people having trouble with is the idea that any _particular_ event in a class of possible events may be rare, but that the likelihood of some event in the class occurring is nearly certain (as Terry Pratchett says: Million to one shots come up nine times out of ten). One sees this a lot in failure analysis. If a system has 100 parts each of which is 95% sure not to fail the chance that some component will fail is over 99%. Or, to put it more colorfully, a lot of girls have relied on the fact that any single act of intercourse is unlikely to result in pregnancy. We call them "mothers". When this is combined with the idea of mean pay out one can quickly see how Lloyds of London sunk so low in such a short time. .- Submitted by Todd Ellner
This is a whole subject of scientific investigation. What kinds of dangers do people routinely over- or underestimate?
It is
not random. A fine example of the Rare Scaremongering is the reaction to the TWA 747 disaster. Airport security
was to be tightened at a cost of billions of dollars--none of which, it appears, would have had the slightest effect
on the disaster. I hope this scheme has been abandoned. - Submitted by Tom Zaslavsky
Ratiocinitis
I'd add: "Does it matter if the store takes the 30% off first? No, not necessarily. In this case, 30% off the original $100 price is $70, and 20% off that gives the same final price, $56 or 44% off. On the other hand, if an absolute amount (instead of a percentage) is offered as a discount as well as a percentage off, the order of operations matters: Suppose the store offers 30% off on everything plus a $10 discount on items with an original price of $100 or more. If we take 30% off the $100 dress and then deduct another $10, the final price is $60 or 40% off; but if we take the $10 off first and then take 30% off the remaining $90, the final price is $63 or 37% off." - Submitted by Steve Tiger
Statistical Brick Wall
My wife was a field biologist until she decided that making a living had its charms. As I understand what she said, the methods for determining numbers of species in an area and numbers of individuals of each species are not perfect, but they are the best measures which we have. The asymptotic method of surveying an area for species has proven out pretty well. Whether or not this can accurately predict the number of species in a larger area is open to some small debate, but it ought not to be dismissed out of hand.- Submitted by Todd Ellner
"Ruling out" all other possible factors: RIGHT ON! This is a typical fallacy, sad to say, of EXPERTS! E.g., in "risk analysis", a dubious "science" that depends on finding all possible factors--among other impossibilities. - Submitted by Tom Zaslavsky
Statistical Rash
The example leaves me perplexed. You say "accident rates" but I think you must mean the percentage
of accidents in which one or both (?) cars were driving at the stated speed. This sounds like a case of insufficient
information--what fallacy is that? By the way, M. Vos Savant is not a reliable authority. Are you familiar with
the case of Vos Savant v. Fermat's Last Theorem?
- Submitted by Tom Zaslavsky
Actually, I am familiar with her criticism of Andrew Wiles proof of Fermat's Last Theorem. I believe at the
time she was just trying to be a skeptical voice during the time between when Wiles first announced his discovery
and when he finally submitted his complete paper. I never believe skepticism in math or science to be a bad thing.
Wiles proof is considered an accepted fact now by the experts. Since I do not know anything about 5 dimension manifolds
and their relationship to primes, I will take their word for it. - Xocxoc
Sucker Bet
I don't know where these two examples fit, but the state of Texas has a relatively new lottery (6 numbers between
1 and 50) and I've heard two things that bother me. One of my friends buys tickets because "someone
has to win". An acquaintance tried to argue that they could never have 1, 2, 3, 4, 5, and 6 occur, because
even though it's supposed to be any six numbers, the population wouldn't believe it and would throw a fit, so the
State wouldn't let that happen. Maybe you can help me with something. I heard someone scoffing over people
who choose their lottery entries from the backs of fortune cookies. Why would choosing them from fortune
cookies be any worse (or better) than choosing them any other way? - Kathryn Donohue
Actually there are a lot of people who think the way you do. In Arizona there was a drawing where the numbers
1, 3, 4, and 6 were all drawn. Since people who match 4 out of 6 win $100, there were a lot of winners from people
betting 1,2,3,4,5,6. If you insist on playing the lottery, some bits of advice: 1) Do not pick the numbers yourself,
let the lottery machine do it for you, this will decrease the chances of having to share the jackpot. 2) If you
are one of those people who like to buy a dollar ticket a week, be aware that the odds of winning one of the 52
lotteries during the course of the year are not as good as the odds of winning if you put all $52 on a single lottery.
3) If you do put all $52 on a single lottery, wait until the jackpot is at least as high as the odds of winning,
in the case of Texas that is $15,890,700. 4) Be aware that the odds of winning a million dollars on a one dollar
investment is much better by playing roulette and winning four times in a row. - Xocxoc
One thing I noted was your calculation for the sucker bet - the calculation appears to be made assuming that
if you win you don't actually pay the $1 cost (or it gets paid back in addition to the $2 million that is the prize
money). Is it just me or do lotteries work just a little differently where you are from ? Not that the 0.00000019
* 1 dollar is going to make a noticeable difference, just the calculation makes that pretty obvious...sorry, couldn't
resist - Submitted by John Chapman
Actually, John, you are correct, the lottery example is a bad example because you lose the dollar even if you
win. The formula is correct for most kinds of wagering - Xocxoc
If I understood this one right, the following might be interesting as additional examples. First, I'd add that the formula will always generate a negative number, because the odds are always against the better--that's how these operations make money. The classic "Numbers Racket" is actually a much better bet than the lottery. Players bet 5 cents ($0.05) on any 3-digit number they choose (anything from 000 to 999). The payoff number is unpredictable, so the chance of picking the right number is 0.001, and the chance of picking a wrong number is 0.999; the payoff on the right number is $25. Applying the formula: [25 x 0.001] - [0.05 x 0.999] = -0.025. Casino roulette is also a better deal than the lottery. A roulette wheel has 36 slots numbered 1 to 36 plus two extra slots called "house numbers" (0 and 00). Then the wheel has a total of 38 slots on which players bet, but the payoff rate is 36 to 1; that is, if you bet $1 on a particular number and that number comes up, you walk away with $36. So the probability that the ball will fall into a particular slot is 1 in 38 (0.0263) and the probability that it will fall into any other slot is 37 in 38 (0.9737) Then the expected return on a $1 bet is [36 x 0.0263] - [1 x 0.9737] = -0.0269, which is much better than we saw with the lottery. (By the way, all the combination bets in roulette are based on the 36-to-1 payoff rate, so the expected return is always the same: a two-number combination pays 18 to 1; three numbers, 12 to 1; four numbers, 9 to 1; six numbers, 6 to 1; twelve numbers, 3 to 1; and eighteen-number combinations--low-vs.-high, odd-vs.-even, red-vs.-black--pay 2 to 1. If you keep betting on some eighteen-number combination, you'll win twice your wager a little less than half the time. The two house-numbers are what gives the edge to the casino.) - Submitted by Steve Tiger
The actual expected return is far lower than you state (at least in most state lotteries). You have to calculate the present value of the jackpot which typically is paid out over 20 years in annual installments. I believe the present value of a 20 year pay out is roughly one third of the stated jackpot value. Factoring in the present value, the expected return on the one dollar ticket in your hypothetical lottery would be closer to 20 cents than to 62 cents. If the lottery takes in $5 million, it can use $700,000 to buy a 20 year $2 million dollar annuity to pay the winner(s), and pocket the remaining $4.3 million. The winners are getting a prize worth $700,000, not $2 million. Vegas started doing this same thing on big slots pay outs a few years ago. What a scam! - Submitted by Paul Hugel
I have a question for you about whether playing a lottery more than once increases your chances of winning during
the period of play. Take a simple example: ten coin-tosses. As you point out on your page, the chances of getting
"heads" or "tails" on each toss is 1:1. This individual chance for each coin toss does not
change if you flip the coin ten times. But how much better than 1:1 are your chances of winning ONCE out of the
ten tosses? How would you calculate that? You mentioned in your response page that your odds are better putting
$52 on one lottery drawing than $1 on 52 lottery drawings. How do you figure the odds of each strategy? Thanks!
- Submitted by Dirk Suringa
The answer is on my new Raw Data page - Xocxoc
Technical Analysis
I found your glossary of math mistakes web page very interesting, but you fell victim to one of the fallacies
yourself when you mentioned technical analysis of stocks. First, there are a lot of folks using it. I'm not sure
why you said there aren't. More to the point, it has lost some of it's luster lately because it can't beat a market
that has been going straight up for 15 years. In the 15 years before that, when the market bounced up to 1000 several
times, but couldn't break above that level, technical analysis produced some very impressive gains - Submitted
by Frank Oslick
I based my statement in the Glossary from reading A Random Walk Down Wall Street by Burton G.
Malkiel which does a pretty convincing job at describing the results of technical analysis. - Xocxoc
The Infamous "Lets Make a Deal Problem"
Your statistical rash example is fine. I see your source was Marily Vos Savant. I only recently
became aware of her when I found some of her old columns on the "Monte Hall" problem. I've searched the
Web and found various places where there are agreements with, and others where there are disagreements with her
position. I am in disagreement. Do you know where I might look to find out first, is she still of the
same opinion, and second, is there a bottom line answer yet, or is this simple situation to be a forever problem.......??????/
- submitted by Chuck Neuenschwander, Dakota County Tech College
For those who are not familiar with it, the "Lets Make a Deal" problem goes as follows:
There are three doors, behind one is a fabulous prize, behind the other two are goats. You pick a door, Monte Hall shows you a door you did not pick which has a goat in it. Now you are given the opportunity to switch to the closed door you did not pick, do you switch?
Marilyn Vos Savant's answer (and mine also) is switch. The best explanation I can give is as follows. There are three possible places the prize can be, just for arguments sake lets say we always start by picking door number 1. There are two possible choices for the second question switch or do not switch. If we do not switch here are the outcomes:
|
Prize is behind |
I pick |
Monty opens |
Result |
|
1 |
1 |
2 or 3 |
Stay on 1, Win |
|
2 |
1 |
3 |
Stay on 1, Lose |
|
3 |
1 |
2 |
Stay on 1, Lose |
|
Prize is behind: |
I pick |
Monty opens |
Result |
|
1 |
1 |
2 |
Switch to 3, Lose |
|
2 |
1 |
3 |
Switch to 2, Win |
|
3 |
1 |
2 |
Switch to 3, Win |
So, if I switch, I will win 2/3rds of the time.
Another Reader Writes:
The Monty hall problem is being looked at incorrectly. there are 24 possible scenarios.
If I switch:
| prize is behind: | I pick: | Monty opens: | result: |
|
1 |
1 |
2 |
switch to 3,lose |
|
1 |
1 |
3 |
switch to 2, lose |
|
2 |
1 |
3 |
switch to 2, win |
|
3 |
1 |
2 |
switch to 3, win |
|
2 |
2 |
1 |
switch to 3,lose |
|
2 |
2 |
3 |
switch to 2, lose |
|
1 |
2 |
3 |
switch to 1, win |
|
3 |
2 |
1 |
switch to 3, win |
|
3 |
3 |
1 |
switch to 2,lose |
|
3 |
3 |
2 |
switch to 1, lose |
|
2 |
3 |
1 |
switch to 2, win |
|
1 |
3 |
2 |
switch to 1, win |
| prize is behind: | I pick: | Monty opens: | result: |
|
1 |
1 |
2 |
win |
|
1 |
1 |
3 |
win |
|
2 |
1 |
3 |
lose |
|
3 |
1 |
2 |
lose |
|
2 |
2 |
1 |
win |
|
2 |
2 |
3 |
win |
|
1 |
2 |
3 |
lose |
|
3 |
2 |
1 |
lose |
|
3 |
3 |
1 |
win |
|
3 |
3 |
2 |
win |
|
2 |
3 |
1 |
lose |
|
1 |
3 |
2 |
lose |
Yet Another Reader Writes:
Your statement about the Monty Hall problem strikes me as a bit deceptive or rather, the problem itself is a
bit deceptive.
The problem with the above tables is that they do not really reflect the true nature of the problem. Since we have
no way of knowing whether our original choice will be the door with the prize, it is useless to base a table on
that possibility. The only thing that we DO know is that Monty will show us one door that has a goat behind it.
Only after that happens does the real decision making begin. The fact that the choice between the two doors is
expressed as "to switch or not to switch" does not change that fact that we are really choosing between
two doors and not three. The third door is present only for dramatic effect (and to satisfy the curiosity of any
audience members who don't know what a goat looks like).
Therefore there is a 50-50 chance of picking the door with the prize and my advice would be to flip a coin. - Submitted
by James Leiser
I do not believe you can view the TWO scenarios involving picking the correct door in the beginning as being equally likely. This is a case of Conditional Probability: A probability in which possible outcomes are eliminated by given information. The chances that it will rain today, and the chances that it will rain today given that there are clouds in the sky, are two different probabilities. The probability of picking the right door at first is 1/3, then the host can pick either of the two doors with a probability of 1/2 for each. So, the probabilities of these are 1/6 each. The probability of picking the wrong door is 2/3, and the host has to show the other wrong door (probability 1) and the prize is behind the door not chosen or shown (also probability 1) Or follow this chart (probabilities in parentheses)
| I Pick | host shows (conditional prob. given my pick) | result by switch (total probability) | ||||
| Door with Prize (1/3) |
|
|
||||
| Wrong Door 1 (1/3) | Wrong Door 2 (1) | win (1/3) | ||||
| Wrong Door 2 (1/3) | Wrong Door 1 (1) | win (1/3) |
And another who gets it right:
I got to the end of the "Reader's Sub." page, and saw the Monte Hall problem, and your example, explaining
why Marilyn Vos Savant is right. One of my stats profs gave a great example of that problem, and how to convince
non-believers (which, according to my prof, includes some of her peers, all with Ph.Ds in math/stats/related fields
- hmmmm...)
In the original problem, you have three doors, you pick one, and Monte shows you which of the other doors is the
wrong choice. The key: he's showing you all but one of the remaining doors, and he'll only show you ones which
are wrong. The best bet is obfuscated by the fact that there're only two other doors - by showing you only
one wrong answer, as that's all that are available without totally giving it away, the magnitude of the new information
gets lost.
Think of this instead: I have a standard deck of poker cards. The winner is the player with the ace of spades.
You get to pick one, face down, and don't get to look at it yet. I have 51 cards left. I thumb through them and
show you 50 of them which aren't the AofS. Now, is the one you picked the AofS (you picked 1 card randomly out
of 52), or is it the one I picked out of the remaining 51 cards (I get to look, and get to pick any of the 51 remaining
cards)? I'd expect you to get the AofS correctly on a single pull, oh... about 1 out of 52 tries. I'd expect me
to have (and hence pick) the AofS about 51 out of 52 times. Put another way: if you managed to get the AofS on
the first pull, I've obviously got to go with something else; otherwise, I'm gonna pick the AofS from what's left,
and win.
Now, let's get in line with the Monte Hall problem - after we both do our picks, you're betting on who's got the
AofS. You can either take the card you picked, or switch and take the one I picked. Go with my choice - it's the
right bet.
I've explained this to others before, and some have said "It isn't the same problem." It IS the same
problem, precisely. It just has the proportions setup so the result is obvious. Reduce the number of cards from
52 to 3, and the correct choice is the same.
Though the original problem IS a good bar bet..... - Submitted by Mike Whalen
Another discussion of this puzzle can be found at http://www.jnlk.com/tiras/doors.htm
This page notes two important assumptions that have to be made: First, the host knows where the prize
is, and second, the host always reveals a goat. If you do not assume these things then your probability of winning
is one in three no matter what you choose.
Another conditional probability problem which is just like this one:
There are two children, at least one is a girl, what is the probability that the other is a girl as well, the surprising
answer is one-third. If there are two children then there are four possibilities: boy-boy, boy-girl, girl-boy,
girl-girl. By saying at least one is a girl we eliminate only the boy-boy choice. In two out of the three possibilities,
the other child is a boy.
Hello, I was at your web page today, which is very interesting by the way, and I was wondering what is wrong with the argument on the home page that says 2 = 1. its so simple but I suppose something must be wrong. thanks. - Submitted by Steve Hill
The whole proof goes as follows:
| Given: | a = b |
| Multiply by a: | a2 = ab |
| Subtract b2 | a2 - b2 = ab - b2 |
| Factor: | (a + b)(a - b) = b(a - b) |
| Divide (a - b): | (a + b) = b |
| Substitute the given: | a + a = a |
| Simplify: | 2a = a |
| Divide by a: | 2 = 1 |
The error is in the fifth step, dividing by (a-b) when a = b means we are dividing by 0, which is a big no-no in mathematics. Try substituting 1 for a and b, Step 4 becomes 2 x 0 = 1 x 0, which is true but it does not mean you can cancel out the 0's.
For more 2 = 1 proofs and their errors, see http://www.math.toronto.edu/mathnet/falseProofs/fallacies.html