The subjective definition of probability seems
to contradict
the aim of physicists to describe the laws of physics
in the most
objective way (whatever this means ).
This is one of the reasons why many regard
the subjective definition of probability
with suspicion (but
probably the main reason is because
we have been taught at university that
``probability is frequency''). The main philosophical
difference between this concept of probability and an
objective definition that
``we would have liked'' (but which does not exist in reality)
is that is not an intrinsic characteristic
of the event , but depends on the state of information
available to whoever evaluates .
The ideal concept of ``objective''
probability is recovered when everybody has the ``same'' state of
information. But even in this case it would be better to speak of
intersubjective probability. The best way to convince
ourselves about
this aspect of probability is to try to ask practical
questions and to evaluate the probability in specific cases,
instead of seeking refuge in abstract questions. I find, in fact,
that -- to paraphrase a famous statement about Time --
``Probability is objective as long as I am not asked to evaluate it.'' Here are
some examples.
Example 1:
``What is the probability
that a molecule of nitrogen at room
temperature has a velocity between 400 and 500 m/s?''. The answer
appears easy: ``take the Maxwell distribution formula from a textbook,
calculate an integral and get a number''. Now
let us change the question:
``I give You
a vessel containing nitrogen
and a detectorcapable of
measuring the speed of a single molecule and You
set up the apparatus (or You let a
person You trust do it).
Now, what is the probability that the
first
molecule that hits the detector has a velocity between
400 and 500 m/s?''. Anybody who has minimal experience (direct
or indirect) of experiments would hesitate before answering.
He would study the problem carefully and perform
preliminary measurements and checks.
Finally he would probably
give not just a single number, but a range of possible numbers
compatible with the formulation of the problem. Then
he starts the experiment and eventually, after 10 measurements,
he may form
a different opinion about the outcome of the eleventh measurement.
Example 2:
``What is the probability that the gravitational constant
has a value between
and
mkgs?''. Before 1994 you
could have looked at the latest
issue of the Particle Data Book[33]
and answered that the probability was . Since then -- as you
probably know -- three new measurements of have been
performed[34]
and we now have four numbers which do not agree
with each other (see Tab. ).
The probability of the true value of
being in that range is currently dramatically decreased.
Table:
Results of measurements of the gravitational constant .
Institute
(ppm)
()
CODATA 1986 (``'')
128
-
PTB (Germany) 1994
83
MSL (New Zealand) 1994
95
Uni-Wuppertal
105
(Germany) 1995
Example 3:
``What is the probability that the mass of the top
quark, or that of any of the supersymmetric particles, is below
20 or
GeV?''. Currently it looks
as if it must be zero. Ten years ago
many experiments were intensively looking
for these particles in those energy ranges.
Because so
many people where searching for them, with
enormous human and capital investment, it meant that,
at that time,
the probability was considered rather high:
high enough for fake signals
to be reported as strong evidence for them3.4.
The above examples show how the evaluation of probability
is conditioned by some a priori (``theoretical'')
prejudices and by some facts (``experimental data''). ``Absolute''
probability makes no sense. Even the classical example
of probability for each of the results in tossing a coin
is only acceptable if: the coin is regular,
it does not remain vertical (not impossible
when playing on the beach),
it does not fall into a manhole, etc.
The subjective point of view is expressed
in a provocative way
by de Finetti's[11]