Re: A question of semantics

Christian Thalmann <cinga@...> wrote:
>
> >     2) The concept of "with probability one", as in statistics.  I
> > have had to try to get this across to people with science degrees
> > and little knowledge of mathematical probability and have had major
> > difficulty.  They often just CAN'T break out of the mindset of
> > discrete mathematics.
>
> I don't quite see the problem with "probability one".  It's
> just a way of saying "with 100% certainty", a concept that
> should be understandable even outside of mathematics.
Yes and no.  Consider a U(0,1) random variable.  The probability
of any particular value is zero - not approximately zero, but
really zero.  And so is the probability of a result being in any
enumerable set.  So, if you take a sample from such a distribution,
the probability of getting those particular values (if you repeated
the sample) is precisely zero.

Many people boggle at the idea of a set of probability one being
less than all possibilities, and yet the omitted elements being
as likely as those in the set of probability one.  And so on.

The concept I am referring to is of something occurring with 100%
certainty and yet not including all possible outcomes.  In any
natural language that I have heard of, there is no distinction
between those two concepts - and the one normally used is that
of including all possible outcomes.

> On the other hand, many people have trouble with applying or
> manipulating non-trivial probabilites, e.g. "if one test has
> a probability of 25% for outcome A, then the probability to
> get an outcome A in four tests is 100%".  ;-)
That is certainly true, but that sort of problem is at a very
different level.


Regards,
Nick Maclaren,
University of Cambridge Computing Service,
New Museums Site, Pembroke Street, Cambridge CB2 3QH, England.
Email:  nmm1@cam.ac.uk
Tel.:  +44 1223 334761    Fax:  +44 1223 334679