http://mathforum.org/kb/thread.jspa?threadID=2222040&tstart=15
Definition of random AP_STAT
Date:
Dec 21, 2010 12:28 PM
Author: Dave Bock
Subject: Re: [ap-stat] definition of random
On Dec 20, 2010, at 10:00 AM, Noelle Shipley wrote:
> We had an interesting conversation in class today about the
definition of "random." Do the outcomes have to be equally likely in
order for an event to be random? Or is an event random merely because
we cannot predict the outcome? In BVD it is implied that the outcomes
must
be
equally
likely,
but
it
does
not
say
explicitly.
It better not!!! In fact, in our teacher supplements we specifically
warn people NOT to let students think random means equally likely. I
think that student confusion sometimes arises when we use equally
likely random digits to simulate random events. The fact that the
random number generator (or table) produces digits 0,1,2,...,9 with a
uniform distribution is what allows us to simulate a rigged coin with a
70% probability of landing tails by saying we'll let the digits 0-6
mean "tails". The coin's behavior is random, but it's the simulation
that's based on equally likely outcomes.
-- Dave
..............................................................
Date: Dec 20, 2010 6:02 PM
Author: dstarnes@lawrenceville.org
Subject: Re: [ap-stat] definition of random
I would say that Random means "due to chance" or "a result of some
chance process", like tossing a coin, rolling a die, drawing a name
from a hat, pulling beads from a bag, using a random number generator,
etc. Random doesn't automatically imply equally likely outcomes. For
instance, if the chance process involves rolling two dice and observing
the sum, the outcomes "7" and "12" are not equally likely.
Daren Starnes
Math Department Chair & Master Teacher
The Lawrenceville School
"Noelle Shipley" noelleeshipley@yahoo.com
12/20/2010 12:12 PM
Please respond to "Noelle Shipley" noelleeshipley@yahoo.com
To"AP Statistics" ap-stat@lyris.collegeboard.com
cc
Subject
[ap-stat] definition of random
We had an interesting conversation in class today about the definition of
"random."
Do the outcomes have to be equally likely in order for an event
to be random?
Or is an event random merely because we cannot predict the
outcome?
In BVD it is implied that the outcomes must be equally likely, but it does
not say explicitly. My gut tells me (and I'm ready to be wrong, or else I wouldn't be
posting here) that randomness has more to do with predictability.
If I have a
rigged coin that has a 70% chance of getting tails, I still don't know
what the outcome will be when I flip it.
The coin is biased, but the flip
is still random.
What say you?
...........................
http://mathforum.org/kb/thread.jspa?threadID=2222241&tstart=15
Date: Dec 20, 2010 12:29 PM
Author: Robert W. Hayden
Subject: [ap-stat] definition of random
Forwarded message:
> From: "Noelle Shipley" noelleeshipley@yahoo.com
>
We
had
an
interesting
conversation
in
class
today
about
the
> definition of "random."
Do the outcomes have to be equally likely
> in order for an event to be random?
Or is an event random merely
> because we cannot predict the outcome?
In BVD it is implied that
> the outcomes must be equally likely, but it does not say explicitly.
> My gut tells me (and I'm ready to be wrong, or else I wouldn't be
>
posting
here)
that
randomness
has
more
to
do
with
predictability.
> If I have a rigged coin that has a 70% chance of getting tails, I
> still don't know what the outcome will be when I flip it.
The coin
>
is
biased,
but
the
flip
is
still
random.
What
say
you?
Funny your should ask!
Like many fundamental concepts, "random" is
not easy to define in words.
Last month I reviewed the classic
_Probability,
Statistics,
and
Truth_
by
Richard
von
Mises.
http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=125
His conclusions are very similar to your own. He attempts a technical
definition of "random" but unless you are up on your metric spaces I am
not sure it will be useful.
I know I am not up on any more recent
attempts.
...............................................
Probability, Statistics, and Truth_ by Richard von Mises.
http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=125
Probability, Statistics, and Truth
Richard von Mises
Publisher: Dover Publications (1981)
Details: 244 pages, Paperback
Edition: 2
Price: $12.95
ISBN: 0486242145
MAA Review
[Reviewed by Robert W. Hayden, on 11/14/2010]
Today this work is generally thought of as the classic statement of the frequentist position on the
foundations of probability as contrasted with a Bayesian approach. In its own day, and even more in the
earlier years of von Mises research, the principal alternative to the frequentist position was not the
Bayesian outlook of the past 50 years but rather the “equiprobable” outlook of many of the pioneers of
probability theory.
These pioneers often tacitly assumed that one could list the possible outcomes and define the probability
of an event as the number of “favorable” outcomes divided by the total number of outcomes. This works
for one toss of one coin, but if you toss two, and list the outcomes as 0, 1 or 2 heads, then the results do
not agree well with the real world behavior of coins. One head comes up about twice as often as none or
two. What von Mises did to deal with this was to assert that probability pertained to limiting values of
relative frequencies of repeated events. This made probability more empirical and less a priori. In
addition, von Mises contributed ideas on determining what situations in the real world might reasonably
be treated via probability. In particular, he had some interesting things to say on the nature of randomness.
Like many of the older books, this one is best understood in the context of its own time and still earlier
times. Transposing it into issues of our own day may be risky. The author does mention subjective
probability here and there, but barely mentions the ideas that today would be labelled “Bayesian.” Today
that approach is most common in business. As an example of the difference in the two approaches,
consider a manager who believes that a new product will appeal to 40% of its market. A survey might
later indicate the appeal is closer to 20%. A frequentist would probably go with the 20% figure while the
Bayesian would try to combine the two estimates into a new and intermediate one.
While the frequentist and Bayesian approaches are often treated as mutually exclusive, it might be more
accurate to say that Bayesians allow frequentist probabilities but allow subjective probabilities as well.
Mathematicians have a very different view of foundations, and some of us might simply say that anything
that satisfies the Kolmogorov axioms is probability, while allowing that some situations are a better match
than others, and that the proof of the pudding should be how well the results of applying probability theory
match reality.
It may also be true that the two approaches suit different problems. A classic 1960 paper by John Tukey
distinguishes between conclusions and decisions. Decisions are unique situations such as our example of
marketing a new product. A decision has to be made once and for all. It that case the Bayesian approach
has appeal. Conclusions are scientific beliefs based on an accumulation of evidence, and in science
including the researcher’s prior opinions is not considered cricket.
It is clear that von Mises’ interest is in the role of probability in science, so though he takes a dim view of
subjective probabilities, he is evaluating them as a basis for scientific conclusions rather than practical
decisions. In light of all this, it might be better to read Tukey and more recent authors on frequentist
versus Bayesian statistics.
One might read von Mises to learn more about the history of probability and statistics and mathematics,
and the connection between these disciplines and science. In its time, the frequentist outlook resolved
some of the paradoxes and inconsistencies in earlier writings. As with any authentic source from an earlier
age, the reader of this book will find some unfamiliar terminology, probably further complicated by the
translation from German. However, for anyone interested in history of the issues it covers, this book is the
book to read.
Reference
Tukey, J. W. (1960) “Conclusions versus Decisions” Technometrics 2 424-433.
-------------------------------------------------------------------------------After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and
universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He
now teaches statistics online at www.statistics.com and does summer workshops for high school teachers
of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics
textbooks to the MAA’s Teaching Statistics.