The lottery and gambling
The United States
government has come to the realization that Japan is leading us in mathematical
literacy. The government's approach to this, as with cigarettes and alcohol, is
to attempt to change our behavior by putting a tax on what they don't like, in
this case mathematical illiteracy. They call this tax the lottery.
Paraphrase of comedian
Emo Phillips
Every American should
learn enough statistics to realize that "One-in-25,000,000" is so
close to "ZERO-in-25,000,000" that not buying a lottery ticket gives
you almost virtually the same chance of winning as when you do buy one!
Mike Snider in MAD
magazine, Super Special December 1995, p.48
I was in college when MacDonald's started their sweepstakes. Finding the correct gamepiece was going to make someone a millionaire. I had a friend named Peter[1] with a hunch. He was going to win.
I was, on the other hand, a math
major. I considered my odds of being that one person in the United States who
would be made incomprehensibly rich. There were a hundred million people trying
to find that one lucky gamepiece. My chances were one in one hundred million of
winning a million dollars. In my book, my long-run expectation was of my winning about a penny. Despite the fact that
I was a poor student, scrounging to find tuition and rent, the prospect of
winning (on average) one cent did not excite me. I was not about to go out of
my way to earn this penny.
I was familiar with the Reader's
Digest Sweepstakes. I had sat down and calculated the expected winnings in the
sweepstakes. I expected to win something less than the price of the postage
stamp I would need to invest in order to submit my entry, so I chose not to
enter.
Peter was not a math major. Peter
knew that if he was to win, he needed to put forth effort to appease the
goddess Tyche[2].
Whenever we went out, whenever we passed the golden arches, he took us through
the drive-through to pick up a gamepiece. Since future millionaires should not
look like tightwads, he would order a little something. He would buy a soda and
maybe an order of fries.
I took Peter to task for his
silly behavior. I explained to him calmly the fundamentals of probability and
expectation. I explained to him excitedly that he was being manipulated, being
duped into spending much more money at MacDonald's than he would have normally.
He told me that he would laugh when he received his one million dollars.
Did he win? No. In college, I
took this as vindication that I was right. This event validated for me a pet
theory: people are not good statisticians[3].
Our state (Wisconsin) has
instituted Emo Phillips' tax on mathematical illiteracy. By not participating,
I am a winner in the lottery. Profits from the lottery go to offset my property
taxes. It is with mixed emotion that I receive this rebate each year. Like
anyone else, I appreciate saving money. I even take a small amount of smug
satisfaction that I win several hundred dollars a year from the lottery, and I
have never purchased a lottery ticket. And I have made this money from people
like Peter, who do not understand statistics.
One newsclip caught me in my
smugness. The report characterized the typical buyer of a lotto ticket as
surviving somewhere near the poverty level. I believe that we all have a right
to decide where to spend our money. I don't think that the government should
only sell lotto tickets to people who can prove that their income is above a certain
level. I am, however, troubled by the image of my taxes being subsidized by an
old woman who is just barely scratching out a living on a pension.
This image was enough for me to
reconsider my mandate that people should base all their decisions on rational
enumeration of the possible outcomes, assignation of probabilities, and
computation of the expectation. What if I were the pensioner who never had
enough money to buy a balanced diet after rent was paid? In the words of the
song, "If you ain't got nothin', you got nothin' to lose." Is the pensioner buying a lottery ticket
because he or she is not capable of rationally considering the options? Or are
all options "bad", so the remote chance of making things
significantly different is worth the risk. Not too long ago, I would have
blamed the popularity of the lottery on mathematical illiteracy. Today I am not
so sure.
Psychological perspective
Where observation is
concerned, chance favors only the prepared mind.
Louis Pasteur
Aristotle maintained
that women have fewer teeth than men; although he was twice married, it never
occurred to him to verify this statement by examining his wives' mouths.
Bertrand Russel, The
Impact of Science on Society
As engineers and scientists, we
like to consider ourselves to be unbiased in our observations of the world.
Worchel and Cooper (authors of the psychology text I learned from) lend support
for our self-evaluation:
[Studies] demonstrate
that if people are given the relevant information, they are capable of
combining it in a logical way...
If we read on, we are given a
different perspective of the ability of the human brain to tabulate statistics:
But will they? ... We
know from studies of memory processes and related cognitive phenomena that
information is not always processed in a way that gives each bit of information
equal access and usefulness.
Worchel and Cooper go on to
describe experimental evidence of people not weighing all data equally.
Furthermore, we tend to be biased in our judgment of an event when we are involved
in that event, our placement of blame in an accident depends on the extent of
damages, and we generally weight a person's behavior higher than we weight the
particular situation the person is in.
The primacy effect
Several other rules can be
invoked to explain our faulty data collection. The first rule to explain what
information is retained is the primacy
effect.
This states that the initial items are more likely to be remembered. This fits
well with folklore like, "You never get a second chance to make a first
impression," and "It is important to get off on the right foot."
Statistically speaking, the primacy effect can be thought of as applying a
higher weighting on the first few data points.
In one experiment of the primacy
effect, the subject is shown a picture of a person, and is given a list
adjectives describing this person. The order of the adjectives is changed for
different subjects. After seeing the picture and word list, the subject is
asked to describe the person. The subject's description most often agrees with
the first few adjectives on the list.
The recency effect
The second rule to explain memory
retention is the recency effect.
This states that, for example, the last items on a list of words (the most
recently seen items) are also more likely than average to be remembered. In
other words, the most recent data points are also more heavily weighted than
average. As an example of this, I remember what I had for lunch today, but I
can barely remember what I had the day before. If my doctor were to ask me what
I normally had for lunch, would my statistics be reliable?
The novelty effect
The third rule states that items or events which are very unusual are apt to be remembered. This is the novelty effect. I once had the pleasure to work in a group with a gentleman who stood 6'5". When he was standing with some other team members who were just over six foot, a remark was made that we certainly had a tall team. In going over the members of this team, I recall four men who were 6'2" or taller. But I also remember a dozen who were an average height of 5'8" to 6', and I recall two others who were around 5'4". The novelty of a man who was seven inches above average, and the image of him standing with other tall men, was enough to substitute for good statistics.
I recall one incident where a
group of engineers was just beginning to get an instrument close to specified
performance. The first time the instrument performed within spec, we joked that
this performance is "typical". The second time the instrument
performed within spec (with many trials in between), we upgraded the level of
performance to "repeatable". The underlying truth of this joking was
the tendency for all of us to only remember those occasions of extremely good
performance.
The paradigm effect
A fourth rule which stands as a
gatekeeper on our memory is the paradigm
effect.
This states that we tend to form opinions based on initial data, and that these
opinions filter further data which we take in. An example of the paradigm
effect will be familiar to anyone who has struggled to debug a computer
program, only to realize (after reading through the code countless times) the
mistake is a simple typographical error. The brain has a paradigm of what the
code is supposed to do. Each time the code is read, the brain will filter the
data which comes in (that is, filter the source code) according to the
paradigm. If the paradigm says that the index variable is initialized at the
beginning, or that a specific line does not have a semi-colon at the end, then
it is very difficult to "see" anything else.
The paradigm effect is more
pervasive than any objective researcher is willing to admit. I have found
myself guilty of paradigms in data collection. I start an experiment with an
expectation of what to see. If the experiment delivers this, I record the
results and carry on with the next experiment. If the experiments fails to
deliver what I expect, then I recheck the apparatus, repeat the calibration,
double check my steps, etc. I have tacitly assumed that results falling out of
my paradigm must be mistakes, and that data which fits my paradigm is correct.
As a result, data which challenges my paradigm is less likely to be admitted
for serious analysis.
An engineer by the name of Harold[4]
had built up some paradigms about the lottery. He showed me that he had
recorded the past few month's of lottery numbers in his computer. He
showed me that three successive lottery numbers had a pattern. When he noticed
this, Harold bought lots of lottery tickets. The pattern unfortunately did not
continue into the fourth set of lottery numbers. As Harold explained it to me,
"The folks at the lottery noticed the pattern and fixed it."
Harold's paradigm was that there
were patterns in the random numbers selected by lottery machines.
Harold had two choices when confronted with a pattern which did not continue
long enough for him to get rich. He could assume that the pattern was just a
coincidence, or he could find an explanation why the pattern changed. In
keeping true to his paradigm, Harold chose the latter. When he explained this
to me, I realized that it was fruitless to try to argue him out of something he
knew to be true. I commented that the folks at the lotto had bigger and faster
computers than Harold, just so they could keep ahead of him.
As another example of the
paradigm effect, consider an engineer named William[5].
William was a heavy smoker and had his first heart attack in his mid-forties.
He was asked once why he kept smoking, when the statistics were so
overwhelming that continuing to smoke would kill him. William replied that his
heart attack was due to stress. Smoking was his way of dealing with stress. To
deprive himself of this stress relief would surely kill him. Furthermore,
stopping smoking is stressful in and of itself.
William's paradigm was that he
was a smoker. No amount of evidence could convince him that this was a bad
idea. Evidently the paradigm is quite strong. In a recent study, roughly half
of bypass patients continue to smoke after the surgery. William had six more
heart attacks and died after his third stroke.
The primacy effect and the paradigm effect working together
The primacy effect and the
paradigm effect often work together to make us all too willing to settle for
inadequate data. My own observation is that people often settle for a few data
points, and are often surprised to find out how shaky their observation is,
statistically speaking.
A case in point is my belief that
young boys are more aggressive than young girls. The first young girls I had
opportunity to closely observe were my own two daughters, who I would not call
aggressive. The first young boy I observed in any detail was the neighbor's,
who I would call aggressive. My
conclusion is that young boys are aggressive, and young girls are not.
Note that, if three people are
picked at random, it is not terribly unlikely that the first person chosen is
aggressive, and the other two are not. In other words, I have no need to appeal
to a correlation between gender and aggressiveness to explain the data. The
simple explanation of chance would suffice.
The primacy effect says that
these three children were the most influential in shaping my initial beliefs.
The paradigm effect says that the future data which I "record" will
be the data which supports my initial paradigm.
In terms of evolution, one would
be tempted to state that an animal with poor statistical abilities would not be
as successful as an animal which was capable of more accurate statistical
analysis. Surely the hypothetical Homo
statistiens
would be able to more accurately assess the odds of finding food or avoiding
predators.
Consider the hypothetical Homo statistiens first encounter with a
saber toothed tiger. Assume that he/she was lucky enough to survive the
encounter. On the second encounter, Homo
statistiens would reason that not enough statistics were collected to
determine whether saber toothed tigers were dangerous. Any good statistician
knows better than to draw any conclusions from the first data point. Clearly,
there is an evolutionary advantage to Homo
sapiens, who jumps to conclusions after the first saber toothed tiger
encounter.
In the words of Desmond Morris,
Traumas... show clearly
that the human animal is capable of a rather special kind of learning, a kind
that is incredibly rapid, difficult to modify, extremely long-lasting and
requires no practice to keep perfect.
The effect of peer pressure
When Richard Feynman was investigating the Challenger disaster, he uncovered another fine example of how poor people are at statistics. He was reading reports and asking questions about the reliability of various components of the Challenger, and found some wild discrepancies in the estimated probabilities of failure. In one meeting at NASA, Feynman asked the three engineers and one manager who were present to write down on a piece of paper the probability of the engine failing. They were not to confer, or to let the others see their estimates. The three engineers gave answers in the range of 1 in 200 to 1 in 300. The manager gave an estimate of 1 in 100,000.
This anecdote illustrates the
wide gap in judgment which Feynman found between management and engineers.
Which estimate is more reasonable? Feynman dug quite deeply into this question.
He talked to people with much experience launching unmanned spacecraft. He
reviewed reports which analytically assessed the probability of failure based
on the probability of failure of each of the subcomponents, and of each of the
subcomponents of the subcomponents, and so on. He concludes:
If a reasonable launch
schedule is to be maintained, engineering often cannot be done fast enough to
keep up with the expectations of the originally conservative certification
criteria designed to guarantee a very safe vehicle... The shuttle therefore
flies in a relatively unsafe condition, with a chance of failure on the order
of a percent.
On the other hand, Feynman is
particularly candid about the "official" probability of failure:
If a guy tells me the
probability of failure is 1 in 105,
I know he's full of crap.
How can it be that the
bureaucratic estimate of failure disagrees so sharply with the more reasonable
engineer's estimate? Feynman speculates that the reason for this is that these
estimates need to be very small in order to ensure continued funding. Would
congress be willing to invest billions of dollars on a program with a one in a
hundred chance of failure? As a result, much lower probabilities are specified,
and calculations are made to justify that this level of safety can be reached.
I am reminded of another
experiment which was devised by the psychologist Solomon Asch in 1951. In this experiment,
the subject was told that this was an experiment investigating perception. The
subject was to sit among four other "subjects", who are actually
confederates. The "subjects" were shown a set of lines on a piece of
paper (for example) and are asked to state out loud which line was longest. The
actors were called on first, one at a time. They were instructed to give
obviously incorrect answers in 12 of 18 trials, but they were to all agree on
the incorrect answer.
It was found in that 75% of
subjects caved into peer pressure, and agreed with the
obviously incorrect answers. When asked about their answers later, away from
the immediate effects of peer pressure, the subjects held to their original
answers, incorrect or not. As far as can be measured with this psychological
experiment, the subjects came to believe that a two inch long line was shorter
than a one inch long line.
So it is with NASA's reliability
data. The data may never have had any shred of credence whatsoever, but simply
by repeating "1 in 100,000" often enough, it became truth.
I have included Feynman's example
not to put down NASA, or promote the ever-popular game of "manager
bashing", but to illustrate this ever-so-human trait that we are all prone
to. We believe what others believe, and we believe what we would like to be
true.
Summary
The effects mentioned here
together support the statement that people do not make good statisticians. The
point which is made is not that "mathematically inept people are poor
statisticians", or that "people are incapable of performing good
statistics". The point is that the natural tendency is for people to not
be good at objectively analyzing data. This goes for high school drop-outs as
well as engineers, scientists and managers. In order for people to produce good
statistics, they need to rely not on their memory and intuition, but on paper
and statistical calculations.
Bibliography
Feynman, Richard P., What do you care what other people
think?, 1988, Penguin Books Canada Ltd.
Flanagan, Dennis, Flanagan's Version, 1989, Random
House
Kresch, David, Crutchfield, Richard S., Livson, Norman, Elements
of Psychology, Third Edition, 1974 Alfred Knopf, Inc.
Morris, Desmond, The Human Zoo, 1969, McGraw Hill
Worchel, Stephen, and Cooper, Joel, Understanding Social
Psychology, revised edition 1979, Doresy Press
[1]
Not his real name.
[2]
Tyche was the Greek goddess of luck.
[3]
I met one person whose behavior indicated that he was not a good statistician,
therefore all people are not good statisticians...[This demonstrates that I am
not a good statistician, since I am content with a sample size of . There are therefore, two people who are not good
statisticians, and this further proves my point.]
[4]
Not his real name, either.
[5]
You guessed it. Not his real name.
