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But I haven't told you a secret.
One of the five million depressors that was inserted into the earth has
red paint daubed on the end of it.
Next, I find an avid lottery
player and I show him the trail of depressors. I tell him that one of
the sticks has red paint on the buried end. If he gives me a dollar and
then pulls up the red-daubed one, I will give him a million dollars. Can
you see him looking away farther than the eye can discern? Can you see
him decide and then say, "What are you trying to tell me? I am to pick
out the one with red paint from those over the whole 79 miles? You must
think I'm crazy."
"No, mister, I don't think
you are crazy. This just shows the chance you take when you invest in
the lottery. Better by far to take the dollar, roll it up and stuff it
in a rat hole. It might choke the rat."
Kenneth Lange
Gardena, California,
USA
E-mail: Kplblange@aol.com
Received June
2, 2000
On Random Musings
In Issue 2 of the Electronic
Journal of Gambling Issues: eGambling: http://www.camh.net/egambling/issue2/research/index.html
Nigel Turner provides an interesting and informative overview
of the nature of randomness and the origins of misunderstandings surrounding
aspects of randomness. There is no doubt that cognitive schemas characterised
by erroneous perceptions, irrational beliefs and distorted cognitions
play a primary role in the maintenance of gambling and problem gambling
behaviours in particular. This view is well articulated in the publications
of key researchers and clinicians such as Robert Ladouceur, Michael Walker
and Tony Toneatto and presented conceptually in the cognitive model offered
by Sharpe and Tarrier. There is no contentious issue for debate within
this context; beliefs are important ingredients fuelling the gambling
urge.
However, on reading Nigel Turner’s
article, I mused over the concept of regression to the mean that was used
to explain why the probability of a toss of coin gradually converged to
a ratio of 50% heads and 50% tails. Turner argues that a difference of
10 heads in a series of 18 tosses is noticeable but that this difference
becomes increasingly negligible with repeated tosses. After a million
tosses, a difference of 10 is so small as to be meaningless. But is this
explanation accurate and valid? Referring to Hayes’ (1969) textbook,
the concept of regression has strong roots in the work of Francis Galton.
Galton noted that in the prediction of natural characteristics there was
an apparent movement to the value of the group average. For example, tall
parents were predicted to have children of smaller height while short
parents were expected to have taller children. Consistent with the linear
prediction rule, it is best-bet practice to predict that an individual
will show a tendency to converge to the group average (regression to the
mean) on any variable chosen. If this were not the case, we would find
a gradual separation of humanity into two classes over generations as
the trend continued for the tall to become taller and the short, very
short. Regression to the mean is not an invariable phenomenon because
exceptions to the rule are possible, tall parents can have taller children.
But stated simply in statistical terms, for a value of any standard score
Zx, the best linear prediction of the standard score Zy
is one relatively nearer the mean of zero than is Zx (Hayes, 1969,
p.500).
In my musings, I wondered whether
the concept of regression to the mean could be validly applied to categorical
random events such as coin tossing, as well as continuous data. Perhaps
the phenomenon of equal probabilities for a heads/tails coin toss, I thought
in this instance, was best explained by recourse to other statistical
laws. By chance, I had recently re-read Wykes (1964) interesting description
of the history of gambling. Contained within its pages was an attempt
to set the reader on the right path to understand why the ratio of heads
to tails in coin tossing approximates 50%. Alan Wykes explains that the
popular view held is that in a series of tosses heads must eventually
come up because of the law of averages. However, he goes on to
state that the phrase ‘law of averages’ is incorrectly used
and in this context is meaningless. What is really meant is the law
of large numbers which states that all cases will happen an equal
number of times as the number of tosses approaches infinity. In
a single toss, the probability of a head is 50%. In the next toss, the
probability remains 50%. The preceding outcome has no influence given
that these tosses are mutually independent events. In a short series of
tosses, it is common for a disproportionate run of outcomes, say heads,
to occur. This is interpreted as the lucky streak by the gambler. But,
as the number of tosses approach infinity, the outcome reveals a 50% probability.
While the end result is similar,
the statistical principles underlying the phenomena of the law of large
numbers and the concept of regression to the mean differ.
References:
- Hayes, W.L. (1969).
- Statistics. New York:
Holt, Rinehart and Winston.
- Wykes, A. (1964).
- The Complete Illustrated
Guide to Gambling. London: Aldus Books.
Alex Blaszczynski,
PhD
Sydney, New South
Wales, Australia
Email: a.blaszczynski@unsw.edu.au
Received:
August 26, 2000
Response to 'On Random Musings'
Regression to the mean is actually
a product of the law of large numbers, so there is no real contradiction.
Regression to the mean in no case requires that the regression will happen.
In fact, if you follow numbers along, then sometimes the percentage of
heads and tails deviate further from 50%, but over the long term will
gradually regress towards 50%. I suppose that to be precise, the law of
large numbers is the principle that explains best what is happening in
this situation of the number of coins, and regressing towards the mean
describes what the percentage is doing — that is getting closer to
50%. Call it what you will, I argue that it is the experience of this
phenomenon, that after such extreme deviations from chance as losing streaks,
that subsequent experience will be more like the norm and give the person
the illusion that the numbers are correcting themselves to conform to
the expected average. Many gamblers call this the law of averages. I call
it regression because it is a regression of the average in one instance
or gambling session to another that produces this illusion. Yes, it is
in fact the law of large numbers operating, with a subsequent sample that
is more like the norm, but it is most likely still a small sample of gambling
experiences.
The use of regression in Galton's
example of the height of different people is also an instance of this
phenomenon. Height is partly determined by chance and it is the presence
of the random component that produces regression over time. If an individual's
score is close to an extreme, the potential range of random deviation
is constrained by the maximum possible range so that the score will most
likely move towards the middle. People in the middle of the distribution
can have children that are either taller or shorter and thus the population's
height remains stable; the number of tall people that have shorter children
is matched by the number of shorter people that have taller children.
Note in fact that Galton's example only really works if there is some
degree of random breeding. Since height is largely determined by genes
and nutrition, you can remove the random component almost completely by
proper nutrition and selective breeding. Great Danes, for example, usually
have offspring that are very similar to their parents, and do not regress
towards the height of the average dog. However, if variation still exists
amongst Great Danes they will regress towards the Great Dane mean. A Great
Dane is a tall dog because its ancestors were selected for their height,
not because of random chance. If random dog breeding were allowed, the
Great Dane offspring would on average be smaller because most other breeds
are smaller.
The following table outlines
the parallel between the height example and gambling sessions to illustrate
why I use the term regression to the mean to describe the experience of
what happens to people.
In the case of height, it is
the random breeding that produces an offspring that is more average. In
the case of gambling sessions after an unusual session of wins or losses,
it is the random wins and losses that produce a session that is more like
the expected average. Of course, by chance the offspring could be as tall
or taller than the parents, and by chance you could have two winning or
two losing sessions in a row. But if chance is operating, the most likely
outcome is that extreme events will be followed by less extreme events.
And I argue that it is the experience of having a great losing streak
(or winning streak) followed by a more average session that produces the
illusion of correction.
As for controversy, I think
there is more controversy than you think. I've talked to numerous people
who believe that solving problem gambling is about helping people deal
with underlying issues, rather than their experiences and beliefs. While
underlying issues are extremely important, I think we need to understand
the beliefs and where they come from in order to solve and prevent problems.
Nigel Turner
Toronto, Ontario,
Canada
E-mail: Nigel_Turner@camh.net]
Received:
August 31, 2000
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